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Pullback attractors for non-autonomous reaction–diffusion equation with infinite delays in $C_{\gamma,L^{r}(\Omega)}$ or $C_{\gamma,W^{1,r}(\Omega)}$

Abstract

In this paper, the well-posedness for the non-autonomous reaction–diffusion equation with infinite delays on a bounded domain is established. The existence of pullback attractors for the process in $C_{\gamma,L^{r}(\Omega)}$ and $C_{\gamma,W^{1,r}(\Omega)}$ is proved, respectively. The noncompact Kuratowski measure is applied to check the asymptotic compactness.

Introduction

Let $\Omega \subseteq \mathbb{R}^{n}$ be a smooth bounded domain. Consider the long-time behavior of the following non-autonomous nonlinear reaction–diffusion equation:

$$ \textstyle\begin{cases} \frac{\partial u}{\partial t}-\Delta u+\lambda u=f(t,u_{t})+g(t,x), & \text{in } [\tau ,+\infty ]\times \Omega , \\ u|_{\partial \Omega} =0, & t>\tau , \\ u(t,x)=\phi (t-\tau ,x),& t\in (-\infty ,\tau ], x\in \Omega , \end{cases} $$
(1)

where $\lambda \geq 0$, and we have the nonlinear term

$$ f\bigl(t,u_{t}(t,x)\bigr)=F\bigl(t,u\bigl(t-\rho (t),x\bigr)\bigr)+ \int^{0}_{-\infty } G\bigl(t,z,u(t+z,x)\bigr)\,dz. $$

Suppose there exist two positive constants $k_{1}$, $k_{2}$, and three positive scalar functions $m_{0}(\cdot )$, $e^{-r\gamma \rho (t)}m _{1}(t)$, $m_{2}(\cdot )e^{-\gamma z}$ which are all in $L^{1}((- \infty ,0],\mathbb{R}^{+})$ such that the functions $F\in C( \mathbb{R}\times \mathbb{R};\mathbb{R})$, $\rho \in C(\mathbb{R};[0,+ \infty ))$, and $G\in C(\mathbb{R} \times (-\infty ,0]\times \mathbb{R};\mathbb{R})$ satisfy

$$\begin{aligned}& \bigl\vert F(t,\upsilon ) \bigr\vert ^{r}\leq \vert k_{1} \vert ^{r}+k_{2}^{r}e^{-r\gamma \rho (t)} \vert \upsilon \vert ^{r}, \quad \forall t,\upsilon \in \mathbb{R}, \end{aligned}$$
(2)
$$\begin{aligned}& \bigl\vert G(t,z,\upsilon ) \bigr\vert \leq m_{0}(z)+m_{1}(z) \vert v \vert , \quad \forall t,\upsilon \in \mathbb{R},z\in (-\infty ,0], \end{aligned}$$
(3)
$$\begin{aligned}& \bigl\vert F(t,\upsilon )-F(t,\nu ) \bigr\vert \leq C_{1}e^{-\gamma \rho (t)} \vert \upsilon - \nu \vert , \quad \forall t, \upsilon ,\nu \in \mathbb{R}, z\in (-\infty ,0], \end{aligned}$$
(4)
$$\begin{aligned}& \bigl\vert G(t,z,\upsilon )-G(t,z,\nu ) \bigr\vert \leq C_{2}m_{2}(z) \vert \upsilon -\nu \vert , \quad \forall t, \upsilon ,\nu \in \mathbb{R},z\in (-\infty ,0], \end{aligned}$$
(5)

and the non-autonomous term $g\in L^{r}_{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))$ ($r>1$) satisfies

$$ \sup_{\tau \leq t}e^{-\delta \tau } \int^{\tau }_{-\infty } \bigl\Vert g(s) \bigr\Vert ^{r}_{X} e^{\delta s}\,ds< \infty ,\quad \forall t\in \mathbb{R}, $$
(6)

for each $\delta \in \{\alpha ,\alpha -L,r(\delta -\eta )\}$, where α, L, δ, η will be given in Lemma 4.1, the local r-power integral is the Bochner integral. We will denote $m_{0}=\int^{0}_{-\infty }m_{0}(s)\,ds$, $m_{1}=\int^{0} _{-\infty }e^{-\gamma s}m_{1}(s)\,ds$, and $m_{2}=\int^{0}_{-\infty }e ^{-\gamma s}m_{2}(s)\,ds$.

Let $C_{\gamma ,X}$ denote the Banach space $C((-\infty ,0];X)$ endowed with the norm

$$ \Vert \phi \Vert _{C_{\gamma ,X}}=\sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert \phi (z) \bigr\Vert _{X}, \quad \gamma >0, $$

where X is $L^{r}{(\Omega )}$ or $W^{1,r}(\Omega )$.

Given $\tau \in \mathbb{R}$, $T>\tau $ and a function $u:(-\infty ,T]\rightarrow X$. For each $t\in [\tau ,T]$, $u_{t}:(- \infty ,0]\rightarrow X$ denotes the function defined by $u_{t}(z)=u(t+z)$ for $z\in (-\infty ,0]$. We are interested in the initial condition $\phi \in C_{\gamma ,X}$.

Retarded differential equations have been used to research many physical systems with non-instant transmission phenomena such as internet data transmission, other memory processes, and specially biological motivations (e.g. species growth or incubating time on disease models [1, 2]). For autonomous systems with delays, the existence of solutions or global attractors has been studied widely in [35] and their qualitative theory has also been well-established. For autonomous systems with variable bounded or unbounded delays, the classical theory extended in [613] has been applied to deal with the existence of solution and special attractors. In fact, autonomous systems with variable delays are non-autonomous in essence. Except that time-periodic equations can be dealt with classic theory relatively straightforward manner, the qualitative properties or asymptotic behavior of many general non-autonomous systems are analyzed by new ideas and methods. In recent years, non-autonomous diffusion equations have attracted much attention in mathematical literature. Duong [14] considered a class of flux-limited diffusions with external force and established the comparison and maximum principles. Jung et al. [15] considered the nonlinear singularly perturbed reaction–diffusion problems in the polygonal domain and proposed a boundary layer analysis which fits a domain with corners.

For the reaction–diffusion systems with finite delays, there are also a sires of work [11, 16, 17]. More recently, Wang et al. [10] proved the existence of pullback attractors in the weighted space $C_{\gamma ,H^{1}(\Omega )}$ for the multi-value process generated by (1) based on the concept of the Kuratowski measure of the noncompactness of a bounded set, where the growth of nonlinear term $F(x,v)$ and $G(x,s,v)$ are both linear, and the non-autonomous term $g(t,x)\in L^{2}_{\mathrm {loc}}(\mathbb{R};L^{2}( \Omega ))$ satisfies

$$ \sup_{\tau \leq t}e^{-\eta \tau } \int^{\tau }_{-\infty } \bigl\Vert g(s) \bigr\Vert ^{2} _{L^{2}{(\Omega )}}e^{\eta s}\,ds< +\infty ,\quad \forall \eta \in \mathbb{R}, \eta >0. $$
(7)

In the present paper, we will prove the existence of solution and the pullback attractors of (1) in the bounded domain of $C_{\gamma , L^{r}{(\Omega )}}$ or $C_{\gamma , W^{1,r}(\Omega )}$ under the conditions (2)–(6) for ${r\geq 2}$.

The main work of this paper contains three issues. Since the space $L^{r}{(\Omega )}$ ($r>2$) loses the inner product and orthogonality, canonical projector and approximation methods [10] are both ineffective to prove the existence of solutions and pullback attractors of (1). In order to overcome this difficulty, we adopt the idea of [17] and decompose (1) into two equations to separate the non-autonomous term to establish well-posedness (see Theorem 3.7 and Theorem 3.10). In addition we investigate the existence of pullback absorbing set by using the approximation technique of [9, 10] to overcome difficulties stemming from infinite delays and infinite dimensions. Consequently, for verifying the asymptotic compactness of (1) in $C_{\gamma ,L^{r}{(\Omega )}}$ ($r>2$), we employ the weak continuous semigroup theory and finite dimensional approximation method in [16, 18] to construct compact embedding results (see Theorem 5.6). Moreover, by improving smooth effect of the semigroup $e^{At}$, we prove the dissipativity and the existence of pullback attractors for (1) in $C_{\gamma ,W^{1,r}(\Omega )}$ (see Lemma 6.1).

The paper is organized as follows. Section 2 gives some preliminaries concerning the definitions of processes and the pullback attractors of non-autonomous dynamical systems. We also give the definition of ω-limit compact and a suitable non-autonomous frameworks for the discussion of attractors in the future. In Sect. 3, we consider the well-posedness of (1) in $C_{\gamma ,L^{r}{(\Omega )}}$ and $C_{W^{1,r}(\Omega )}$, respectively. In Sects. 4 and 6, we prove the existence of bounded absorbing sets in both spaces above. In Sects. 5 and 7, the existence of pullback attractors in $C_{\gamma ,L^{r}{( \Omega )}}$ and $C_{\gamma ,W^{1,r}(\Omega )}$ is proved.

Preliminaries

Let X be a complete metric space with metric $d_{X}(\cdot ,\cdot )$. Denote by $H^{*}_{X}(\cdot ,\cdot )$ the Hausdorff semi-distance between two nonempty subsets of a complete metric space X, which is defined by

$$ H^{*}_{X}(A,B)=\sup_{a\in A}\inf _{b\in B}d_{X}(a,b). $$

Definition 2.1

A mapping $U(t,\tau ):X\rightarrow X$, $t> \tau $ in $\mathbb{R}$, is called a process if

  1. (1)

    $U(\tau ,\tau )x=x$, $\forall \tau \in \mathbb{R}$, $x\in X$;

  2. (2)

    $U(t,s)U(s,\tau )x=U(t,\tau )x$, $\forall \tau \leq s\leq t \in \mathbb{R}$, $x\in X$.

Definition 2.2

The Kuratowski measure $k(A)$ of noncompactness of the set A is defined by

$$ k(A)=\inf\{\delta >0\mid A \text{ admits a finite cover by sets whose diameter} \leq \delta \}. $$

Definition 2.3

Let $\{U(t,\tau )\}$ be a process on X. We say that $\{U(t,\tau )\}$ is

  1. (1)

    pullback dissipative, if there exists a family of bounded sets $\mathcal{D}=\{{D(t)}\}_{t\in \mathbb{R}}$ in X so that, for any bounded set $B\subset X$ and each $t\in \mathbb{R}$, there exists a $S_{0}=S_{0}(B,t)\in \mathbb{R}^{+}$ such that

    $$ U(t,t-s)B\subset D(t), \quad \forall s\geq S_{0}; $$
  2. (2)

    $\mathcal{{D}}$-pullback ω-limit compact with respect to each $t\in \mathbb{R}$, if, for any $\varepsilon >0$, there exists a $S_{1}=S_{1}(\mathcal{{D}},t,\varepsilon )\in \mathbb{R}^{+}$ such that

    $$ k \biggl(\bigcup_{s\geq S_{1}}U(t,t-s)D(t-s) \biggr)\leq \varepsilon . $$

Proposition 2.4

If the process $\{U(t,\tau )\}$ is $\mathcal{D}$-pullback ω-limit compact in X, then $\{U(t,\tau )\}$ is pullback ω-limit compact for any bounded subset B of X.

It follows from Theorem 3 of [10].

Definition 2.5

A family of nonempty compact subsets $A=\{A(t)\}_{t\in \mathbb{R}}$ of X is called to be a pullback attractor for the process $\{U(t,\tau )\}$ if

  1. (1)

    $\mathcal{A}=\{A(t)\}_{t\in \mathbb{R}}$ is invariant, i.e.,

    $$ U(t,\tau )A(t)=A(t), \quad \forall t\geq \tau ,\tau \in \mathbb{R} ; $$
  2. (2)

    $\mathcal{A}$ is pullback attracting, i.e., for every bounded set B of X and any fixed $t\in \mathbb{R}$,

    $$ \lim_{s\rightarrow +\infty } H^{*}_{X} \bigl(U(t,t-s)B,A(t)\bigr)=0. $$

Definition 2.6

Let $\{U(t,\tau )\}$ be a process on X. We say that ${U(t,\tau )\zeta }$ is norm-to-weak continuous in ζ for any fixed $t\geq \tau $, $\tau \in \mathbb{R}$, if there exists a sequence $\zeta_{n}\rightarrow \zeta $ in X and $t_{n} \rightarrow t$ such that ${U(t_{n},\tau )\zeta_{n}}\rightharpoonup {U(t,\tau )\zeta }$ (weak convergence).

The general existence of pullback attractors has been given as follows [10].

Proposition 2.7

Let X be a Banach space, and let $\{U(t,\tau )\}$ be a process on X. Let $U(t,\tau ) \zeta $ is norm-to-weak continuous in x for fixed $t\geq \tau $, $\tau \in {\mathbb{R}}$. If, for any fixed $t\in \mathbb{R}$, $\forall T\in \mathbb{R}^{+}$, $\bigcup_{t\geq T}D(t)$ is bounded, the process $\{U(t,\tau )\}$ is pullback dissipative and $\mathcal{{D}}$-pullback ω-limit compact with respect to each $t\in \mathbb{R}$, then $\{U(t,\tau )\}$ possesses a pullback attractor in $\mathcal{A}=\{A(t)\}_{t\in \mathbb{R}}$ in X given by

$$ A(t)=\bigcap_{T\in \mathbb{R}^{+}} \overline{{\bigcup _{s\geq T}U(t,t-s)D(t-s)}}\subset D(t). $$

Existence of solutions

By a solution $u\in C((-\infty ,T];X^{1})$ of (1), we mean that, for any $T>0$, $z\in (-\infty ,0]$, $\tau < t\leq T$,

$$\begin{aligned} u(t) =&e^{\Delta (t-\tau )}u(\tau )+ \int^{t}_{\tau }e^{\Delta (t-s)}\bigl[- \lambda u+f(x,u_{s})+g(x,s)\bigr]\,ds, \\ =&e^{\Delta (t-\tau )}u(\tau )+ \int^{t}_{\tau }e^{\Delta (t-s)}\bigl[- \lambda u +f \bigl(x,u{(s+z)}\bigr)+g(x,s)\bigr]\,ds, \end{aligned}$$
(8)

where $u(t)=\phi (t-\tau ,x)$, $u(\tau )=\phi (0,x)$, $t\in (-\infty , \tau ]$.

Let $A=\Delta $. $X^{\alpha }$ is the fractional power space associated to the operator Δ. The linear operator $A=\Delta $ with Dirichlet boundary conditions in a bounded and smooth domain Ω can be seen as an unbounded operator in $L^{r}(\Omega )$, $1< r< \infty $, with domain $D(A)=W^{2,r}(\Omega )\cap W^{1,r}_{0}(\Omega )$. In this situation, $-A=-\Delta $ is a sectorial operator and generates an analytic semigroup $e^{At}$ in $L^{r}({\Omega })$. Denote by $\{E^{\alpha }_{r}\}_{\alpha \in \mathbb{R}}$ the fractional power spaces associated to A with the norm $\Vert u \Vert _{E^{\alpha }_{r}}= \Vert (-A)^{ \alpha }u \Vert _{L^{r}({\Omega })}$, $u\in E^{\alpha }_{r}$. Notice that $E^{0}_{r}=L^{r}(\Omega )$ and $E^{1}_{r}=W^{2,r}(\Omega )\cap W^{1,r} _{0}(\Omega )$. It follows from [19] that the semigroup $e^{At}$ has the following smooth effect:

$$\begin{aligned} \bigl\Vert e^{At}x \bigr\Vert _{E^{\beta }_{r}}\leq t^{-(\beta -\alpha )} \Vert x \Vert _{E^{ \alpha }_{r}},\quad x\in E^{\beta }_{r}, t>0, 0\leq \alpha \leq \beta . \end{aligned}$$
(9)

Since the embedding $E^{1}_{r}\hookrightarrow E^{0}_{r}$ is compact, we know from Remark 6.1 of [20] that the resolvent of −A is compact, and the embedding $E^{\alpha }_{r}\hookrightarrow E^{\beta } _{r}$ is continuous and compact for $\forall \alpha >\beta $.

Local existence of solutions for (1) in $C_{\gamma,L^{r}(\Omega)}$ ($1< r<\infty$)

In order to apply Theorem 1 [18] to prove the existence of a solution for (1), we decompose system (1) into a linear system and a non-autonomous nonlinear system as follows, respectively:

$$ \textstyle\begin{cases} \frac{\partial v}{\partial t}-\Delta v =g(t,x) & \text{in } [\tau ,+\infty ]\times \Omega , \\ v|_{\partial \Omega} =0, & t>\tau , \\ v(t,x)=0,& \tau \in \mathbb{R}, t\in (-\infty ,\tau ], x\in \Omega , \end{cases} $$
(10)

and

$$ \textstyle\begin{cases} \frac{\partial w}{\partial t}-\Delta w=\tilde{f}(x,w_{t})+f_{1}(w) & \text{in } [\tau ,+\infty ]\times \Omega , \\ w|_{\partial \Omega} =0, & t>\tau , \\ w(t,x)=\phi (t-\tau ,x),& \tau \in \mathbb{R}, t\in (-\infty ,\tau ], x\in \Omega , \end{cases} $$
(11)

where $\tilde{f}(x,w_{t})=f(x,w_{t}+v_{t})$, $f_{1}(w)=-\lambda (w+v)$, $u _{t}=v_{t}+w_{t}$.

Lemma 3.1

([21])

For any $\tau \leq t_{1}< t_{2}$, $\frac{1}{p}+\frac{1}{q}=1$,

$$\begin{aligned} \biggl\Vert \int^{t_{2}}_{t_{1}}e^{A(t_{2}-s)}g(x,s)\,ds \biggr\Vert _{L^{r}( \Omega )}\leq \bigl\Vert g(x,t) \bigr\Vert _{L^{p}_{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))} (t _{2}-t_{1})^{\frac{1}{q}}. \end{aligned}$$

Furthermore, Eq. (10) has a unique solution $v(t)$ in the sense of (8) such that

$$\begin{aligned} v(t)\in C\bigl([\tau ,T_{0}+\tau ];L^{r}(\Omega )\bigr) \end{aligned}$$

satisfies

$$\begin{aligned} v(t)= \int^{t}_{\tau }e^{A(t-s)}g(x,s)\,ds, \end{aligned}$$
(12)

where $T_{0}$ is chosen in Lemma 3.6 later.

Proof

$$\begin{aligned}& \biggl\Vert \int^{t_{2}}_{t_{1}}e^{A(t_{2}-s)}g(x,s)\,ds \biggr\Vert _{L^{r}( \Omega )} \\& \quad \leq \int^{t_{2}}_{t_{1}} \bigl\Vert g(x,t) \bigr\Vert _{L^{p}_{\mathrm {loc}}(\mathbb{R};L^{r}( \Omega ))}\,ds \\& \quad \leq \biggl( \int^{t_{2}}_{t_{1}} \,ds \biggr)^{\frac{1}{q}} \biggl( \int^{t_{2}}_{t_{1}} \bigl\Vert g(x,t) \bigr\Vert _{L^{p}_{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))}\,ds \biggr)^{\frac{1}{p}} \\& \quad \leq \bigl\Vert g(x,t) \bigr\Vert _{L^{p}_{\mathrm {loc}}({t_{1}}, {t_{2}};L^{r}(\Omega ))} ( {t_{2}}-{t_{1}})^{\frac{1}{q}}. \end{aligned}$$

Note that we can choose $0<{t_{2}}-{t_{1}}\leq 1$. □

Lemma 3.2

Assuming (2)(5) hold, we have

$$\begin{aligned}& \bigl\Vert \tilde{f}(t,w_{t})+f_{1}(w)\bigr\Vert _{X^{1} }\leq C_{3}(\lambda +1) \bigl(1+ \Vert w_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}} \bigr), \end{aligned}$$
(13)
$$\begin{aligned}& \bigl\Vert \tilde{f}(t,w_{t})-\tilde{f}(t, \nu_{t})+f_{1}(w)-f_{1}(\nu ) \bigr\Vert _{X ^{1}}\leq C_{4}(\lambda +1) \Vert w_{t}- \nu_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}, \end{aligned}$$
(14)

where $w,\nu \in C((-\infty ,T_{0}+\tau ];L^{r}(\Omega ))$, $t\in (\tau ,T_{0}+\tau ]$.

Proof

Denote $X^{\alpha }_{r}:=E^{\alpha -1}_{r}$, $\alpha \in \mathbb{R}$. Especially, $X^{1}_{r}:=L^{r}(\Omega )$. For any $u,\psi \in C((-\infty ,T_{0}+\tau ];L^{r}(\Omega ))$ and any $t\in (\tau ,T_{0}+\tau ]$ we get

$$\begin{aligned} \bigl\Vert F(t,u_{t}) \bigr\Vert _{X^{1}} \leq & C_{5}\bigl( \bigl\Vert k_{1}+k_{2}e^{-\gamma \rho (t)}u _{t} \bigr\Vert _{X^{1}}\bigr) \\ \leq & C_{5}\bigl(k_{1} \vert \Omega \vert +k_{2} \Vert u_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\bigr) \\ \leq & C_{5} \bigl(1+ \Vert u_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\bigr) \end{aligned}$$
(15)

and

$$\begin{aligned}& \biggl\Vert \int^{0}_{-\infty }G\bigl(t,z,u(t+z)\bigr) \,dz \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq \biggl\Vert \int^{0}_{-\infty }\bigl( \bigl\vert m_{0}(z) \bigr\vert +m_{1}(z) \bigl\vert u(t+z) \bigr\vert \bigr)\,dz \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq m_{0} \vert \Omega \vert +m_{1} \Vert u_{t} \Vert _{C_{\gamma ,L^{r}( \Omega )}} \\& \quad \leq C_{6} \bigl(1+ \Vert u_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}} \bigr). \end{aligned}$$
(16)

Combining with (15) and (16), for any $u,\psi \in C(( \tau ,T_{0}+\tau ];X^{1 })$, we have

$$\begin{aligned} \bigl\Vert f(t,u_{t}) \bigr\Vert _{X^{1}} \leq & C_{3} \bigl(1+ \Vert u_{t} \Vert _{C_{\gamma ,L ^{r}(\Omega )}} \bigr). \end{aligned}$$
(17)

By (4) and (5), we find

$$\begin{aligned}& \bigl\Vert f(t,u_{t})-f(t,\psi_{t}) \bigr\Vert _{X^{1}} \\& \quad \leq C_{1}e^{-\gamma \rho (t)} \bigl\Vert u\bigl(t-\rho (t)\bigr)-v \bigl(t-\rho (t)\bigr) \bigr\Vert _{L ^{r}(\Omega )}+ C_{2} \biggl\Vert \int^{0}_{-\infty }m_{1}(z) \vert u_{t}-\psi_{t} \vert \,dz \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq C_{3} \Vert u_{t}-\psi_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}, \end{aligned}$$
(18)

where $C_{3}$ and $C_{4}$ depend on $(k_{1},k_{2},m_{0},m_{1},m_{2})$. From (17) and (18), we obtain

$$\begin{aligned} \bigl\Vert \tilde{f}(t,w_{t}) \bigr\Vert _{{X^{1}}} \leq & C'_{3} \bigl( \Vert w_{t} \Vert _{C_{ \gamma ,L^{r}(\Omega )}}+1\bigr), \end{aligned}$$
(19)

and

$$\begin{aligned} \bigl\Vert \tilde{f}(t,w_{t})-\tilde{f}(t, \nu_{t}) \bigr\Vert _{L^{r}(\Omega )} =& \bigl\Vert f(t,w _{t}+\nu_{t})-f(t,w_{t}+\nu_{t}) \bigr\Vert _{L^{r}(\Omega )} \\ \leq & C'_{4} \Vert w_{t}- \nu_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$
(20)

Hence, (13) and (14) are obvious. □

Lemma 3.3

If $u\in C((-\infty ,T_{0}+\tau ],L^{r}(\Omega ))$, then, for all $t\in (\tau ,T_{0}+\tau ]$, $z\in (-\infty ,0]$, we have

$$\begin{aligned} \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\bigl(f_{1}(w)+ \tilde{f}(t,w_{s})\bigr)\,ds \biggr\Vert _{L^{r}(\Omega )}\leq C( \lambda +1) (t-\tau ) \bigl(\omega (t)+1\bigr), \end{aligned}$$
(21)

where

$$\begin{aligned} \omega (t)=\Bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + \sup _{\theta \in (\tau ,t]} \bigl\Vert w(\theta )+v(\theta ) \bigr\Vert _{{L ^{r}(\Omega )}}\Bigr). \end{aligned}$$

Proof

By (9), it is not difficult to see that

$$\begin{aligned}& \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\tilde{f}(t,w_{s})\,ds \biggr\Vert _{L ^{r}(\Omega )} \\& \quad \leq C(\lambda +1) \int^{t}_{\tau } \bigl(1+ \Vert w_{s}+v_{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\bigr)\,ds \\& \quad \leq C (\lambda +1) \int^{t}_{\tau } \Bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}( \Omega )}} + \sup_{\theta \in (\tau ,s]} \bigl\Vert w(\theta )+v(\theta ) \bigr\Vert _{{L^{r}(\Omega )}}\Bigr)\,ds +C(\lambda +1) (t-\tau ) \\& \quad \leq C(\lambda +1) (t-\tau )\omega (t)+C(\lambda +1) (t-\tau ). \end{aligned}$$
(22)

 □

Lemma 3.4

For any $t\in (\tau ,T_{0}+\tau ]$, $z\in (-\infty ,0]$ and any $w,\nu \in C((-\infty ,T_{0}+\tau ],L^{r}(\Omega ))$ be such that $(t-\tau ) \Vert w_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\leq \mu $, $(t-\tau ) \Vert \nu_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\leq \mu $, for some $\mu >0$. Then we have

$$\begin{aligned}& \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\bigl[\bigl( \tilde{f}(s,w_{s})-\tilde{f}(s, \nu_{s})\bigr)+ \bigl(f_{1}\bigl(w(s)\bigr)- f_{1}\bigl(\nu (s)\bigr)\bigr) \bigr]\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq C(1+\lambda ) (t-\tau ) \sup_{\theta \in (\tau ,t]} \bigl\Vert w( \theta )-\nu (\theta ) \bigr\Vert _{L^{r}(\Omega )} . \end{aligned}$$
(23)

Proof

$$\begin{aligned}& \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\bigl[\bigl( \tilde{f}(s,w_{s})-\tilde{f}(s, \nu_{s})\bigr)+ \bigl(f_{1}\bigl(w(s)\bigr)- f_{1}\bigl(\nu (s)\bigr)\bigr) \bigr]\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq C (1+\lambda ) \int^{t}_{\tau } \Vert w_{s}- \nu_{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\,ds \\& \quad \leq C (1+\lambda ) (t-\tau ) \sup_{\theta \in (\tau ,t]} \bigl\Vert w( \theta ) -\nu (\theta ) \bigr\Vert _{ { L^{r}(\Omega )}}. \end{aligned}$$
(24)

 □

Lemma 3.5

([22])

Assume $u:(-\infty ,T_{0})\rightarrow X$ is continuous and $u_{\tau }=\phi $. If there exists a nondecreasing function $m(t)\geq 0$ such that

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{X}\leq \bigl\Vert \phi (\tau ) \bigr\Vert _{X}+m(t), \quad \textit{for all } -\infty < t\leq T_{0}, \end{aligned}$$

then

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u(t+z) \bigr\Vert _{X}\leq \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert \phi (t+z) \bigr\Vert _{X}+m(t),\quad -\infty < t\leq T_{0}. \end{aligned}$$
(25)

Lemma 3.6

Assume (2)(6) hold. Let $1< r<\infty $, $z\in (- \infty ,0]$. For any $\chi_{\tau }\in C( (-\infty ,0]; L^{r}(\Omega ))$, there exist $R(\chi_{\tau })>0$ and $T_{0}=T_{0}(\chi_{\tau })$ with the property that, for any $\phi \in B_{C_{\gamma ,L^{r}(\Omega )}}( \chi_{\tau },R)$, there exists a continuous function $w(\cdot ;\phi (0))$ with $w_{\tau }=\phi $:

$$\begin{aligned} w\in C\bigl([\tau ,T_{0}+\tau ];L^{r}(\Omega )\bigr) \end{aligned}$$
(26)

such that, for any $t\in [\tau ,T_{0}+\tau ]$, w is the unique solution of Eq. (11) in the sense of (8). This solution is a classical solution and for any $t\in (\tau ,T_{0}+ \tau ]$, satisfies

$$\begin{aligned} w_{t}\in C\bigl((-\infty ,0];L^{r}(\Omega ) \bigr) \end{aligned}$$
(27)

and

$$\begin{aligned} \lim_{t\rightarrow \tau^{+}}(t-\tau )\sup_{z\in (-\infty ,0]}e^{ \gamma z} \bigl\Vert w(t+z,\phi ) \bigr\Vert _{L^{r}(\Omega )}=0, \end{aligned}$$
(28)

and, moreover, if $\phi_{1},\phi_{2}\in B_{C_{\gamma ,L^{r}(\Omega )}}( \chi_{\tau },R)$ then

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert w(t+z, \phi_{1})-\nu (t+z,\phi _{2}) \bigr\Vert _{L^{r}(\Omega )} \leq M_{1}e^{M_{2}(t-\tau )} \Vert \phi_{1}-\phi _{2} \Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$
(29)

Furthermore, the time of existence is uniform on any bounded set (respectively, compact set) S of $C_{\gamma ,L^{r}(\Omega )}$.

Proof

Fix $\mu >0$ and for any $\tau \in \mathbb{R}$, $\forall t\in (-\infty ,\tau ]$, let $\Vert \phi \Vert _{C_{\gamma ,L^{r}( \Omega )}}\leq \mu $. We will use the contraction mapping principle to establish the existence of a solution for (11).

Let

$$\begin{aligned} K(T_{0})=\Bigl\{ w\in C\bigl((-\infty ,T_{0}+\tau ];L^{r}(\Omega )\bigr),t\in (\tau ,T _{0}+\tau ]: \sup _{t\in (\tau ,T_{0}+\tau ]} \bigl\Vert w(t) \bigr\Vert _{ { L^{r}(\Omega )}}\leq \mu +1\Bigr\} , \end{aligned}$$

with the norm

$$\begin{aligned} \Vert w \Vert _{K(T_{0})}=\sup_{t\in (\tau ,T_{0}+\tau ]} \bigl\Vert w(t) \bigr\Vert _{{L^{r}(\Omega )}}, \end{aligned}$$

where $T_{0}$ is determined later. So that $(K, \Vert \cdot \Vert )$ is a nonempty complete metric space. For each $t\in (\tau ,T_{0}+\tau ]$, we introduce the mapping

$$\begin{aligned} \begin{aligned} & \Phi :K(T_{0})\rightarrow C\bigl((-\infty ,T_{0}+\tau ];X^{1}\bigr), \\ &\Phi (w) (t)= \textstyle\begin{cases} e^{\Delta (t-\tau )}w(\tau )+\int^{t}_{\tau }e^{\Delta (t-s)}[f_{1}(w)+ \tilde{f}(s,w_{s})]\,ds, &t> \tau , \\ w(t,x)=\phi (t-\tau ,x),&t\in (-\infty ,\tau ]. \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(30)

Let us first prove that Φ is a well-defined map and $\Phi (K(T _{0}))\subset K(T_{0})$. We start by showing that

$$\begin{aligned} \text{if } w \in K(T_{0}), \text{then } \Phi (w) \in C\bigl( (-\infty ,T_{0}+\tau ];L^{r}(\Omega )\bigr). \end{aligned}$$
(31)

Fixing $t_{2}\in (\tau ,T_{0}+\tau ]$, and letting $T_{0}+ \tau \geq t_{1}>t_{2}$, then we have

$$\begin{aligned}& \bigl\Vert (\Phi w) (t_{1} )-(\Phi w) (t_{2}) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl\Vert \bigl( e^{-A(t_{1})}-e^{-A(t_{2})}\bigr)w(\tau ) \bigr\Vert _{L^{r}( \Omega )} + \biggl\Vert \int^{t_{1}}_{t_{2}}e^{A(t_{1}-s)}\tilde{f}(s,w_{s})\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad\quad {} + \biggl\Vert \int^{t_{1}}_{t_{2}}e^{A(t_{1}-s)}f_{1} \bigl(w(s)\bigr)\,ds \biggr\Vert _{L ^{r}(\Omega )} + \biggl\Vert \bigl[I-e^{-A(t_{1}-t_{2})}\bigr] \int^{t_{2}}_{\tau }e ^{A(t_{2}-s)} \tilde{f}(s,w_{s})\,ds \biggr\Vert _{L^{r}(\Omega )}. \end{aligned}$$

In the above, the first and fourth term trivially go to zero as $t_{1}\rightarrow t_{2}$. Let us consider the second term. For this term we have

$$\begin{aligned}& \biggl\Vert \int^{t_{1}}_{t_{2}}e^{A(t_{1}-s)}\tilde{f}(s,w_{s})\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq C \int^{t_{1}}_{t_{2}} \bigl(1+ \Vert w_{s}+v_{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\bigr)\,ds \\& \quad \leq C\Bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}}+ \sup_{s\in (\tau ,t_{1}]} \bigl\Vert w(s)+v(s) \bigr\Vert _{ { L^{r}(\Omega )}}\Bigr) ( {t_{1}}-{t_{2}}) +C({t_{1}}-{t_{2}}) \\& \quad \leq C\omega (t) ({t_{1}}-{t_{2}}) +C({t_{1}}-{t_{2}}), \end{aligned}$$

which goes to zero as $t_{1}\rightarrow t_{2}^{+}$. Similarly, the third term also goes to zero as $t_{1}\rightarrow t_{2}^{+}$. The case $t_{1}< t_{2}$ is similar.

Let us now show that $\Vert \Phi (w)(t) \Vert _{L^{r}(\Omega )}\leq \mu +1$, for all $t\in (\tau ,T_{0}+\tau ]$. For $\chi_{\tau } \in C((-\infty ,0]; L^{r}(\Omega ))$ fixed, choose $r\ll 1$ and $T_{0}\leq \frac{1-r}{C(\lambda +1)(1+\omega (t))}$ such that, for any $t\in (\tau ,T_{0}+\tau ]$, by (9), we have $\Vert e^{ A(t- \tau )}\chi_{\tau } \Vert _{L^{r}(\Omega )}\leq \mu $, and $\Vert e ^{ A(t-\tau )}r \Vert _{L^{r}(\Omega )}\leq r$.

Based on the above fact, we have

$$\begin{aligned}& \bigl\Vert \Phi (w) (t) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl\Vert e^{-A(t-\tau )}w(\tau ) \bigr\Vert _{L^{r}(\Omega )} +C ( \lambda +1) (t-\tau ) +C(1+\lambda ) (t-\tau ) \int^{t}_{\tau } \Vert w _{s} \Vert _{C_{L^{r}(\Omega )}}\,ds \\& \quad \leq \bigl\Vert e^{-A(t-\tau )}r \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}}+ \bigl\Vert e^{-A(t-\tau )}\chi_{\tau } \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}} +C ( \lambda +1) (t-\tau ) \bigl(1+\omega (t)\bigr) \\& \quad \leq r+ \Vert \chi_{\tau } \Vert _{C_{\gamma ,L^{r}(\Omega )}} +C ( \lambda +1) (t-\tau ) \bigl(1+\omega (t)\bigr) \\& \quad \leq \mu +r+C (\lambda +1) (t-\tau ) \bigl(1+\omega (t)\bigr). \end{aligned}$$

On the other hand, it follows from Lemma 3.3 that Φ is a strict contraction in $K(T_{0})$ and that

$$\begin{aligned} \bigl\Vert \Phi (w)-\Phi (\nu ) \bigr\Vert _{K(T_{0})}\leq C(\lambda +1) (t-\tau )\omega (t) \Vert w- \nu \Vert _{K(T_{0})}, \quad t\in [\tau ,T_{0}+\tau ]. \end{aligned}$$

The simple computations above suggest that we can choose $T_{0}$ small enough so that the map Φ is contraction from $K(T_{0})$ into itself. By the Banach contraction principle we see that Φ has a unique fixed point in $K(T_{0})$. We will denote this fixed point by $w(t,\phi )$ for $t\in (\tau ,T_{0}+\tau ]$, $\phi \in C((-\infty ,0],L ^{r}(\Omega ))$, and it is defined for $\Vert \phi -\chi_{\tau } \Vert _{C_{ \gamma ,L^{r}(\Omega )}}\leq \rho$. Note that from (31) $w(t,\phi )\in C((-\infty ,T_{0}+\tau ];L^{r}(\Omega ))$.

Let us prove that $(t-\tau ) \Vert w_{t} \Vert _{C_{\gamma ,L ^{r}(\Omega )}} \rightarrow 0$ as $t\rightarrow \tau^{+}$.

From Lemma 3.3,

$$\begin{aligned}& (t-\tau ) \bigl\Vert w(t) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq (t-\tau ) \bigl\Vert e^{A(t-\tau )}\phi (0) \bigr\Vert _{L^{r}(\Omega )} + (t-\tau ) \int^{t}_{\tau } \bigl\Vert e^{A(t-s)} \bigl(f_{1}(w)+\tilde{f}(s,w _{s})\bigr) \bigr\Vert _{L^{r}(\Omega )}\,ds \\& \quad \leq (t-\tau ) \bigl\Vert \phi (0) \bigr\Vert _{L^{r}(\Omega )} +C(1+ \lambda ) (t-\tau ) \int^{t}_{\tau }\bigl(1+ \Vert w_{s} \Vert _{C_{\gamma ,L^{r}( \Omega )}}\bigr)\,ds \\& \quad\quad {} +C(1+\lambda ) (t-\tau ) \Vert v_{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$

By Lemma 3.5, we obtain

$$\begin{aligned}& (t-\tau ) \Vert w_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}} \\& \quad \leq (t-\tau ) \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}}+C (1+ \lambda ) (t-\tau ) \int^{t}_{\tau } \Vert w_{s} \Vert _{C_{\gamma ,L ^{r}(\Omega )}}+C(1+\lambda ) (t-\tau ). \end{aligned}$$

Thus by the Gronwall inequality, we have

$$\begin{aligned}& (t-\tau ) \Vert w_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}} \\& \quad \leq (t-\tau ) \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + C (1+ \lambda ) (t-\tau ) \\& \quad\quad {} + \bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + C(1+ \lambda ) \bigr) (t-\tau )C(1+\lambda ) \int^{t}_{\tau }\exp {\bigl(C (1+ \lambda )\bigr) (t-s)}\,ds \\& \quad \leq \bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + C (1+ \lambda ) \bigr) (t-\tau ) \\& \quad\quad {} +C(1+\lambda ) \bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + C(1+\lambda ) \bigr) (t-\tau )^{2}\exp {\bigl(C(1+\lambda ) (t-\tau )\bigr)} \overset{t\rightarrow \tau ^{+}}{\rightarrow }0. \end{aligned}$$

Moreover, if $\forall \phi_{1},\phi_{2}\in B_{{C_{\gamma ,L ^{r}(\Omega )}} }(\chi_{\tau },r)$, taking into account the estimates of Lemma 3.3 and our choice of $T_{0}$, we have

$$\begin{aligned}& \bigl\Vert w\bigl(t,\phi_{1}(0)\bigr)-\nu \bigl(t, \phi_{2}(0)\bigr) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl\Vert e^{A(t-\tau )}\bigl(\phi_{1}(0)- \phi_{2}(0)\bigr) \bigr\Vert _{L^{r}( \Omega )} \\& \quad\quad {} + \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\bigl[ \tilde{f}(s,w_{s})-\tilde{f}(s, \nu_{s})+f_{1}(w)-f_{1}( \nu )\bigr]\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl\Vert (\phi_{1}-\phi_{2}) \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}} +C(1+\lambda ) \int^{t}_{\tau } \Vert w_{s}- \nu_{s} \Vert _{C_{L^{r}( \Omega )}}\,ds \\& \quad \leq \bigl\Vert (\phi_{1}-\phi_{2}) \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}} +C(1+\lambda ) (t-\tau ) \bigl\Vert (\phi_{1}- \phi_{2}) \bigr\Vert _{C_{\gamma ,L ^{r}(\Omega )}} \\& \quad\quad {} +C(1+\lambda ) \int^{t}_{\tau } \sup_{\theta \in (\tau ,s]} \bigl\Vert w( \theta )-\nu (\theta ) \bigr\Vert _{{L^{r}(\Omega )}}\,ds. \end{aligned}$$

By Lemma 3.5, we have

$$\begin{aligned}& \sup_{\theta \in (\tau ,t]} \bigl\Vert w\bigl(t, \phi_{1}(0)\bigr)-\nu \bigl(t,\phi_{2}(0)\bigr) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl(1+C(1+\lambda ) (t-\tau )\bigr)e^{C(1+\lambda )(t-\tau )} \bigl\Vert ( \phi_{1}-\phi_{2}) \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$

Furthermore,

$$\begin{aligned}& \bigl\Vert w_{t}(\cdot,\phi_{1})-\nu_{t}(\cdot, \phi_{2}) \bigr\Vert _{C_{\gamma ,L ^{r}(\Omega )}} \\& \quad \leq \bigl(1+C(1+\lambda ) (t-\tau )\bigr) \bigl\Vert (\phi_{1}- \phi_{2}) \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}}e^{C(1+\lambda )(t-\tau )} \\& \quad \leq M_{1}(t-\tau ) \bigl\Vert (\phi_{1}- \phi_{2}) \bigr\Vert _{C_{\gamma ,L ^{r}(\Omega )}}e^{M_{1}(t-\tau )}, \end{aligned}$$

where $M_{1}=1+C(1+\lambda )$.

This concludes the existence of the theorem. Notice that, from the existence part, we see that, for any $\phi \in B_{{C_{\gamma ,L ^{r}(\Omega )}} }(\chi_{\tau },R)$, there exists a unique solution in the sense of (8), defined in $[\tau ,T_{0}+\tau ]$. The uniqueness of solutions for Eq. (11) is proved. □

Theorem 3.7

Assume (2)(6) hold. Let $1< r<\infty $, $g\in L^{r} _{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))$ ($r>1$), $z\in (-\infty ,0]$. If $\nu_{\tau }\in C((-\infty ,0];L^{r}(\Omega ))$, there exist $0< R(\nu_{\tau })\leq R(\chi_{\tau })$ and $T_{0}(\nu_{\tau })\leq {T_{0}(\chi_{\tau })} $ with the property that, for any $\phi \in B _{C_{\gamma ,L^{r}(\Omega )}}(\nu_{\tau },R) $, there exists a continuous function $u(\cdot ;\phi (0))$ with $u_{\tau }=\phi $:

$$\begin{aligned} u\in C\bigl([\tau ,T_{0}+\tau ];L^{r}(\Omega )\bigr), \end{aligned}$$
(32)

which is the unique solution of (1) in the sense of (8). This solution is a classical solution and $\forall t \in (\tau ,T_{0}+\tau ]$ it satisfies

$$\begin{aligned} u_{t}\in C\bigl((-\infty ,0];L^{r}(\Omega ) \bigr) \end{aligned}$$
(33)

and

$$\begin{aligned} \lim_{t\rightarrow \tau^{+}}(t-\tau )\sup_{z\in (-\infty ,0]}e^{ \gamma z} \bigl\Vert u( t+z,\phi ) \bigr\Vert _{L^{r}(\Omega )}=0; \end{aligned}$$
(34)

if $\forall \phi_{1},\phi_{2}\in B_{\gamma ,L^{r}(\Omega )}( \upsilon_{\tau },r)$, then

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u_{1}(t+z,\phi_{1})-u_{2}(t+z, \phi_{2}) \bigr\Vert _{L^{r}(\Omega )}\leq M_{1}(t-\tau )e^{M_{1}(t-\tau )} \Vert \phi_{1}-\phi_{2} \Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$
(35)

Furthermore, the time of existence is uniform on any bounded set (respectively, compact set) S of $C_{\gamma ,L^{r}(\Omega )}$.

Proof

By Lemma 3.1 and Lemma 3.6, Eq. (1) has a unique solution $u\in C((-\infty ,T_{0}];L^{r}(\Omega ))$ satisfying (33)–(35). □

Local existence of solutions of (1) in $C_{\gamma,W^{1,r}(\Omega)}$ ($1< r< N$)

Lemma 3.8

([21])

For any $t_{1}< t_{2}$, $0< \frac{1}{q}-\frac{1}{2}$, where $\frac{1}{r}+\frac{1}{q}=1$, we have

$$\begin{aligned}& \biggl\Vert \int^{t_{2}}_{t_{1}}e^{A(t_{2}-s)}g(x,s)\,ds \biggr\Vert _{W^{1,r}( \Omega )} \\& \quad \leq \biggl(\frac{1}{1-\frac{q}{2}} \biggr)^{\frac{1}{q}} \bigl\Vert g(x,t) \bigr\Vert _{L^{r}_{b}(t_{1},t_{2};L^{r}{(\Omega )})} (t_{2}-t_{1})^{ \frac{1}{q}-\frac{1}{2}}. \end{aligned}$$

Furthermore, Eq. (10) has a unique solution $v(t)$ in the sense of (8) such that

$$\begin{aligned} v(t)\in C\bigl([\tau ,T_{0}];W^{1,r}(\Omega )\bigr)\cap C \bigl([\tau ,T_{0}+\tau ];W ^{2,r}(\Omega )\bigr) \end{aligned}$$

satisfies

$$\begin{aligned} v(t)= \int^{t}_{\tau }e^{A(t-s)}g(x,s)\,ds. \end{aligned}$$
(36)

Proof

We have

$$\begin{aligned}& \biggl\Vert \int^{t_{2}}_{t_{1}}e^{A(t_{2}-s)}g(x,s)\,ds \biggr\Vert _{W^{1,r}( \Omega )} \\& \quad \leq \biggl\Vert \int^{t_{2}}_{t_{1}}(t_{2}-s)^{-\frac{1}{2}}g(x,s)\,ds \biggr\Vert _{L^{r}{(\Omega )}} \\& \quad \leq \biggl( \int^{t_{2}}_{t_{1}}(t_{2}-s)^{-\frac{q}{2}}\,ds \biggr)^{\frac{1}{q}} \biggl( \int^{t_{2}}_{t_{1}} \bigl\Vert g(x,s) \bigr\Vert ^{r}_{L^{r}_{b}(\mathbb{R};L^{r}{( \Omega )})}\,ds\biggr)^{\frac{1}{r}} \\& \quad \leq \biggl(\frac{1}{1-\frac{q}{2}} \biggr)^{\frac{1}{q}} \bigl\Vert g(x,t) \bigr\Vert _{L^{r}_{\mathrm {loc}}(t_{1},t_{2};L^{r}{(\Omega )})} (t_{2}-t_{1})^{ \frac{1}{q}-\frac{1}{2}}. \end{aligned}$$

 □

Lemma 3.9

Assume (2)(6) hold. Let $1< r< N$, $z\in (-\infty ,0]$. If $\chi_{\tau }\in C((-\infty ,0]; W^{1,r}(\Omega ))$, there exist $R(\chi_{\tau })>0$ and $T_{0}( \chi_{\tau })>0 $ with the property that $\forall t\in (-\infty , \tau )$ for any $\phi \in B_{C_{\gamma ,W^{1,r}(\Omega )}}(\chi_{ \tau },R)$, there exists a continuous function $w(\cdot ;\phi (0))$ with $w_{\tau }=\phi $:

$$\begin{aligned} w\in C\bigl([\tau ,T_{0}+\tau ];W^{1,r}( \Omega )\bigr), \end{aligned}$$
(37)

which is the unique solution of (11) in the sense of (8). This solution is a classical solution and $\forall t \in (\tau ,T_{0}+\tau ]$, $z\in (-\infty ,0]$, satisfies

$$\begin{aligned} w_{t}\in C\bigl((-\infty ,0];W^{1,r}(\Omega ) \bigr) \end{aligned}$$
(38)

and

$$\begin{aligned} \lim_{t\rightarrow \tau^{+}}(t-\tau )\sup_{z\in (-\infty ,0]}e^{ \gamma z} \bigl\Vert w(t+z,\phi ) \bigr\Vert _{W^{1,r}(\Omega )}=0, \end{aligned}$$
(39)

and if $\phi_{1},\phi_{2}\in B_{C_{\gamma ,W^{1,r}(\Omega )}}(\chi _{\tau },R)$, then

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert w(t+z, \phi_{1})-\nu (t+z,\phi _{2}) \bigr\Vert _{W^{1,r}(\Omega )} \leq M_{1}T_{0}e^{M_{1}(t-\tau )} \Vert \phi _{1}- \phi_{2} \Vert _{C_{\gamma ,W^{1,r}(\Omega )}}. \end{aligned}$$
(40)

Furthermore, the time of existence is uniform on any bounded set (respectively, compact set) S of $C_{\gamma ,W^{1,r}(\Omega )}$.

Proof

For $\forall t\in (\tau ,T_{0}+\tau ]$, $z\in (-\infty ,0]$ and any $w,\nu \in C((-\infty ,T_{0}+\tau ];W^{1,r}(\Omega ))$, using (2),(3), we obtain (13) and (14). The remaining part of the proof is similar to Lemma 3.6. □

Theorem 3.10

Assume (2)(6) hold. Let $1< r<\infty $, $r>1$, $z\in (-\infty ,0]$. If $\nu_{\tau }\in C((-\infty ,0]; W^{1,r}(\Omega ))$, there exist $0< R(\nu_{\tau })\leq R(\chi_{\tau })$ and $T_{0}(\nu_{\tau })\leq T_{0}(\chi_{\tau })$ with the property that for any $\phi \in B_{C_{\gamma ,W^{1,r}(\Omega )}}(\nu_{\tau },R)$, there exists a continuous function $u(\cdot ;\phi (0))$ with $u_{\tau }= \phi $:

$$\begin{aligned} u\in C\bigl([\tau ,T_{0}+\tau ];W^{1,r}( \Omega )\bigr), \end{aligned}$$
(41)

which is the unique solution of (11) in the sense of (8). This solution is a classical solution and $\forall t \in [\tau ,T_{0}+\tau ]$ it satisfies

$$\begin{aligned} u_{t}\in C\bigl((-\infty ,0];W^{1,r}(\Omega ) \bigr), \quad\quad \lim_{t\rightarrow \tau ^{+}}(t-\tau ) \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u(t+z,\phi ) \bigr\Vert _{W^{1,r}(\Omega )}=0, \end{aligned}$$
(42)

and if $\phi_{1},\phi_{2}\in B_{C_{\gamma ,W^{1,r}(\Omega )}}(\nu_{ \tau },R) $, then

$$\begin{aligned} \begin{aligned}[b] &\sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u(t+z, \phi_{1})-u(t+z,\phi_{2}) \bigr\Vert _{W^{1,r}(\Omega )}\\ &\quad \leq M_{1}(t-\tau )e^{M_{1}(t-\tau )} \Vert \phi_{1}- \phi_{2} \Vert _{C_{\gamma ,W^{1,r}(\Omega )}}. \end{aligned} \end{aligned}$$
(43)

Furthermore, the time of existence is uniform on any bounded set (respectively, compact set) S of $C_{\gamma ,W^{1,r}(\Omega )}$.

Proof

It follows from Lemmas 3.8 and 3.9. The proof is similar to Theorem 3.7. Here we omit the details. □

Uniform estimates in $C_{\gamma,L^{r}(\Omega)}$

Lemma 4.1

Assume that (2), (3), and (6) hold, $g\in L^{r}_{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))$, and there exists a positive constant α such that

$$\begin{aligned} \bigl(\lambda -(\varepsilon_{2}+ {m}_{1} + \varepsilon_{4}) ({r-1})- \alpha \bigr)>0 \end{aligned}$$
(44)

and

$$\begin{aligned} L:= \biggl({m}_{1}+\frac{2^{r} k_{2}^{r}}{\lambda^{(r-1)}} \biggr)< \alpha \leq r\gamma . \end{aligned}$$
(45)

Then, for any initial data $\phi \in C_{\gamma ,L^{r}(\Omega )}$, any solution $u_{t}$ of Eq. (1) satisfies

$$\begin{aligned} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \leq & r e^{\alpha \tau }e ^{-\alpha t} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+ \frac{ \alpha }{\alpha -L}C_{\Omega }+ \varepsilon_{4}^{-(r-1)}e^{-\alpha t} \int^{t}_{-\infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds \\ & {} +r e^{(\alpha -L)\tau }e^{(L-\alpha )t} \Vert \phi \Vert ^{r}_{C_{\gamma , {L^{r}(\Omega )}}} \\ & {} + \varepsilon_{4}^{-(r-1)}e^{(L-\alpha )t} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r} \bigr)\,ds, \end{aligned}$$
(46)

where $\varepsilon_{2}$, $\varepsilon_{4}$ will be determined later on.

Proof

Multiplying (1) by $\vert u(t) \vert ^{r-2}u(t)$ and integrating by parts, we get

$$\begin{aligned}& \frac{1}{r}\frac{d}{dt} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )}+\frac{4(r-1)}{r ^{2}} \int_{\Omega } \bigl\vert \nabla \bigl( \bigl\vert u(t) \bigr\vert ^{\frac{r}{2}}\bigr) \bigr\vert ^{2}\,dx+ \int_{ \Omega } \lambda \bigl\vert u(t) \bigr\vert ^{r}\,dx \\& \quad = \int_{\Omega } F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,dx + \int_{ \Omega } \int^{0}_{-\infty }G\bigl(t,s,u(t+s)\bigr) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,ds\,dx \\& \quad\quad {} + \int_{\Omega } g(t,x) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,dx. \end{aligned}$$
(47)

We fix two positive parameters $\varepsilon_{1}$ and $\varepsilon_{4}$ that will be chosen later. Then, by assumptions (2), (6) and Young’s inequality, we have

$$\begin{aligned}& \int_{\Omega }F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) \vert u \vert ^{r-2}u\,dx \\& \quad \leq \int_{\Omega } \bigl\vert F\bigl(t,u\bigl(x,t-\rho (t)\bigr) \bigr) \bigr\vert \bigl\vert u(t) \bigr\vert ^{(r-1)}\,dx \\& \quad \leq \frac{2^{r}\varepsilon_{1}^{-(r-1)}}{r} \vert k_{1} \vert ^{r} \vert \Omega \vert ^{r} +\frac{2^{r}\varepsilon_{1}^{-(r-1)}}{r}k_{2}^{r} \Vert u_{t} \Vert ^{r}_{C _{\gamma ,L^{r}(\Omega )}} + \varepsilon_{1}\biggl(\frac{r-1}{r}\biggr) \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \end{aligned}$$
(48)

and

$$\begin{aligned} \int_{\Omega }g(t,x) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,dx \leq & \int_{\Omega } \bigl\vert g(t,x) \bigr\vert \bigl\vert u(t) \bigr\vert ^{(r-1)}\,dx \\ \leq &\frac{\varepsilon_{4}^{-(r-1)}}{r} \bigl\Vert g(t) \bigr\Vert _{L^{r}(\Omega )} ^{r} +\varepsilon_{4} \biggl(\frac{r-1}{r} \biggr) \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}( \Omega )}. \end{aligned}$$
(49)

Therefore

$$\begin{aligned}& \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )}+\frac{4(r-1)}{r} \int_{\Omega } \bigl\vert \nabla \bigl( \bigl\vert u(t) \bigr\vert ^{\frac{r}{2}}\bigr) \bigr\vert ^{2}\,dx +\bigl(r\lambda -( \varepsilon_{1}+\varepsilon_{4}) ({r-1})\bigr) \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \,dx \\& \quad \leq \varepsilon_{1}^{-(r-1)} \bigl(k_{1} \vert \Omega \vert ^{r} +k_{2}^{r} \Vert u _{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \bigr) +r \int_{\Omega } \int^{0} _{-\infty }G\bigl(t,s,u(t+s)\bigr) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,ds\,dx \\& \quad\quad {} +\varepsilon_{4}^{-(r-1)} \bigl\Vert g(t) \bigr\Vert _{L^{r}(\Omega )}^{r}. \end{aligned}$$
(50)

Let $\alpha >0$, it will also be determined later. Then

$$\begin{aligned}& \frac{d}{dt} \bigl(e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \bigr) \\& \quad = \alpha e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} +e^{\alpha t} \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq -\frac{4(r-1)}{r}e^{\alpha t} \int_{\Omega } \bigl\vert \nabla \bigl( \bigl\vert u(t) \bigr\vert ^{ \frac{r}{2}}\bigr) \bigr\vert ^{2}\,dx -\bigl(r\lambda -( \varepsilon_{1}+\varepsilon_{4}) ( {r-1})-\alpha \bigr)e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad\quad {} + \varepsilon_{1}^{-(r-1)}e^{\alpha t} \vert k_{1} \vert ^{r} \vert \Omega \vert ^{r}+ \varepsilon_{1}^{-(r-1)}e^{\alpha t}k_{2}^{r} \Vert u_{t} \Vert ^{r}_{C_{ \gamma ,L^{r}(\Omega )}}+ \varepsilon_{4}^{-(r-1)}e^{\alpha t} \bigl\Vert g(t) \bigr\Vert _{L^{r}(\Omega )}^{r} \\& \quad\quad {} +re^{\alpha t} \int_{\Omega } \int^{0}_{-\infty }G\bigl(t,s,u(t+s)\bigr) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,ds\,dx. \end{aligned}$$
(51)

Integrating from τ to t, we have

$$\begin{aligned}& e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq e^{\alpha \tau } \bigl\Vert u(\tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} - \int^{t} _{\tau }\bigl(r\lambda -(\varepsilon_{1}+ \varepsilon_{4}) ({r-1})-\alpha \bigr)e ^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )} \,dx \\& \quad\quad {} + \varepsilon_{1}^{-(r-1)} \vert k_{1} \vert ^{r} \vert \Omega \vert ^{r}\frac{e^{\alpha t}}{ \alpha }+ \varepsilon_{1}^{-(r-1)}k_{2}^{r} \int^{t}_{\tau }e^{\alpha s} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}}\,ds \\& \quad\quad {} +r \int^{t}_{\tau }e^{\alpha s} \int_{\Omega } \int^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr) \bigl\vert u(s) \bigr\vert ^{r-2}u(s)\,dz\,dx\,ds \\& \quad\quad {} +\varepsilon_{4}^{-(r-1)} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L ^{r}(\Omega )}^{r}\,ds. \end{aligned}$$
(52)

By assumption (3), (6) and Young’s inequality, we obtain

$$\begin{aligned}& r \biggl\vert \int^{t}_{\tau }e^{\alpha s} \int_{\Omega } \int^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr) \bigl\vert u(s) \bigr\vert ^{r-2}u(s)\,dz\,dx\,ds \biggr\vert \\& \quad \leq r \int^{t}_{\tau }e^{\alpha s} \int_{\Omega } \int^{0}_{-\infty } \bigl\vert G\bigl(s,z,u(s+z)\bigr) \bigr\vert \bigl\vert u(s) \bigr\vert ^{r-1}\,dz\,dx\,ds \\& \quad \leq \varepsilon_{2}^{-(r-1)}m_{0}^{r} \vert \Omega \vert ^{r} \int^{t}_{ \tau }e^{\alpha s}\,ds + \varepsilon_{2}(r-1) \int^{t}_{\tau }e^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )}\,ds \\& \quad \quad {} +\varepsilon_{3}^{-(r-1)} {m}_{1} \int^{t}_{\tau }e^{\alpha s} \Vert u _{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}^{r}\,ds +\varepsilon_{3}(r-1) {m} _{1} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )}\,ds \\& \quad \leq \varepsilon_{2}^{-(r-1)}m_{0}^{r} \Vert \Omega \Vert ^{r}_{L^{r}( \Omega )}\frac{e^{\alpha t}}{\alpha } + \varepsilon_{2}(r-1) \int^{t} _{\tau }e^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )}\,ds \\& \quad\quad {} +\varepsilon_{3}^{-(r-1)} {m}_{1} \int^{t}_{\tau }e^{\alpha s} \Vert u _{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}^{r}\,ds +\varepsilon_{3}(r-1) {m} _{1} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )}\,ds, \end{aligned}$$
(53)

where $\varepsilon_{2}$ and $\varepsilon_{3}$ are other positive constants to be determined later.

Combining (52)–(53) we conclude that

$$\begin{aligned}& e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq e^{\alpha \tau } \bigl\Vert u(\tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} + \biggl(\frac{ k_{1} \vert \Omega \vert ^{r}}{\varepsilon_{1}^{(r-1)}\alpha }+ \frac{m_{0}^{r} \vert \Omega \vert ^{r}}{\varepsilon_{2}^{(r-1)}\alpha } \biggr)e^{\alpha t} \\& \quad\quad {} - \bigl(r\lambda -(\varepsilon_{1}+\varepsilon_{2}+ \varepsilon_{3} {m}_{1} +\varepsilon_{4}) ({r-1})-\alpha \bigr) \int^{t}_{\tau }e^{ \alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )} \,ds \\& \quad\quad {} + \biggl(\frac{{m}_{1}}{\varepsilon_{3}^{(r-1)}}+\frac{ k_{2}^{r}}{ \varepsilon_{1}^{(r-1)}} \biggr) \int^{t}_{\tau } e^{\alpha s} \Vert u_{s} \Vert _{C_{\gamma ,L^{r}({\Omega })}}^{r}\,ds + \frac{1}{\varepsilon_{4}^{(r-1)}} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds . \end{aligned}$$
(54)

Choosing $\varepsilon_{1}=\lambda $, $\varepsilon_{3}=1$, we now can choose positive constants $\varepsilon_{2}$ and $\varepsilon_{4}$ small enough such that $(\lambda -(\varepsilon_{2}+\bar{m}_{1} +\varepsilon _{4})({r-1})-\alpha )>0$. Then

$$\begin{aligned}& e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq e^{\alpha \tau } \bigl\Vert u(\tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} + \biggl(\frac{ k_{1} \vert \Omega \vert ^{r}}{\lambda^{(r-1)}\alpha } + \frac{m_{0}^{r} \vert \Omega \vert ^{r}}{\varepsilon_{2}^{(r-1)}\alpha } \biggr)e^{\alpha t} \\& \quad\quad {} + \biggl({m}_{1}+\frac{ k_{2}^{r}}{\lambda^{(r-1)}} \biggr) \int^{t}_{ \tau }e^{\alpha s} \Vert u_{s} \Vert _{C_{L^{r}({\Omega })}}^{r}\,ds+\varepsilon _{4}^{-(r-1)} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )} ^{r}\,ds . \end{aligned}$$
(55)

Now set $t+\theta $ instead of t, where $\theta \in (-\infty ,0]$. By the assumption (45), we have $\alpha \leq r\gamma $. Multiplying (55) by $e^{-\alpha ( t+\theta )}$ and $e^{r\gamma \theta } e^{-r\gamma \theta }$, it follows that

$$\begin{aligned} \sup_{\theta \in (\tau -t,0]}e^{r\gamma \theta } \bigl\Vert u(t+ \theta ) \bigr\Vert ^{r} _{L^{r}(\Omega )} \leq & e^{-\alpha t}e^{\alpha \tau } \Vert \phi \Vert ^{r} _{C_{\gamma ,{L^{r}(\Omega )}}}+ C_{\Omega }+ \frac{e^{-\alpha t}}{ \varepsilon_{4}^{(r-1)}} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega )}^{r}\,ds \\ & {} + \biggl({m}_{1}+\frac{ k_{2}^{r}}{\lambda^{(r-1)}} \biggr)e^{-\alpha t} \int^{t}_{\tau }e^{\alpha s} \Vert u_{s} \Vert _{C_{\gamma ,L^{r}({\Omega })}} ^{r}\,ds, \end{aligned}$$
(56)

where

$$\begin{aligned} C_{\Omega }= \biggl(\frac{ k_{1} \vert \Omega \vert ^{r}}{\lambda^{(r-1)}\alpha } + \frac{m_{0}^{r} \vert \Omega \vert ^{r}}{\varepsilon_{2}^{(r-1)}\alpha } \biggr). \end{aligned}$$
(57)

Note that

$$\begin{aligned} e^{r\gamma \theta } \bigl\Vert u(t+\theta ) \bigr\Vert ^{r}_{L^{r}(\Omega )} =&e^{r \gamma \theta } \bigl\Vert \phi (t+\theta -\tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} =e ^{-r\gamma (t-\tau )}e^{r\gamma (t+\theta -\tau )} \bigl\Vert \phi (t+\theta - \tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\ \leq & e^{-r\gamma (t-\tau )} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}( \Omega )}}} \leq e^{-\alpha (t-\tau )} \Vert \phi \Vert ^{r}_{C_{\gamma , {L^{r}(\Omega )}}}, \quad \forall \theta \in (-\infty ,\tau -t]. \end{aligned}$$

Let $L:={m}_{1}+\frac{2^{r} k_{2}^{r}}{\lambda^{(r-1)}}<\alpha $. Then it yields

$$\begin{aligned} e^{\alpha t} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}} \leq & re ^{\alpha \tau } \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+C_{\Omega } e^{\alpha t}+ \varepsilon_{4}^{-(r-1)} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds \\ & {} +\biggl({m}_{1}+\frac{2^{r} k_{2}^{r}}{\lambda^{(r-1)}}\biggr) \int^{t}_{\tau }e ^{\alpha s} \Vert u_{s} \Vert _{C_{\gamma ,L^{r}({\Omega })}}^{r}\,ds \\ \leq &re^{\alpha \tau } \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+C _{\Omega } e^{\alpha t}+ \varepsilon_{4}^{-(r-1)} \int^{t}_{\tau }e ^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds \\ & {} +L \int^{t}_{\tau }e^{\alpha s} \Vert u_{s} \Vert _{C_{\gamma ,L^{r}({\Omega })}} ^{r}\,ds. \end{aligned}$$

By Fubini’s theorem and Grownwall’s lemma, we find that

$$\begin{aligned} e^{\alpha t} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}} \leq &r e ^{\alpha \tau } \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+ \varepsilon_{4}^{-(r-1)} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega )}^{r}\,ds \\ & {} +r e^{(\alpha -L)\tau }e^{Lt} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}( \Omega )}}} +\frac{\alpha }{\alpha -L}C_{\Omega }e^{\alpha t} \\ & {} + \varepsilon_{4}^{-(r-1)}e^{Lt} \int^{t}_{\tau } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r} \bigr)\,ds. \end{aligned}$$
(58)

Hence, (6) and condition (45) imply that

$$\begin{aligned} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \leq & Cr e^{-\alpha t} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+ \frac{\alpha }{\alpha -L}C _{\Omega }+ \varepsilon_{4}^{-(r-1)}e^{-\alpha t} \int^{t}_{-\infty }e ^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds \\ & {} +r e^{(\alpha -L)\tau }e^{(L-\alpha )t} \Vert \phi \Vert ^{r}_{C_{\gamma , {L^{r}(\Omega )}}} \\ & {} + \varepsilon_{4}^{-(r-1)}e^{(L-\alpha )t} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r} \bigr)\,ds \\ \overset{\vartriangle }{=}&R_{1,C_{\gamma ,L^{r}(\Omega )}} (t, \phi ,g,\alpha ,L). \end{aligned}$$
(59)

For each $t\in \mathbb{R}$, let

$$\begin{aligned} B_{R_{1,C_{\gamma ,L^{r}(\Omega )}}}(t)=\bigl\{ u\in {C_{\gamma ,L^{r}( \Omega )}}\mid \Vert u \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}}\leq R_{1,C_{\gamma ,L^{r}( \Omega )}}(t,\phi ,g,\alpha ,L)\bigr\} , \end{aligned}$$
(60)

which implies that the family of bounded sets $B=\{B_{R_{1,C_{\gamma ,L^{r}(\Omega )}}}(t)\}_{t\in \mathbb{R}}$ is pullback absorbing for the process $\{U(t,\tau )\}$ on $C_{\gamma ,L^{r}(\Omega )}$. □

Existence of the pullback attractors in $C_{\gamma,L^{r}(\Omega)}$ ($r>2$)

In this section, we will discuss the case where the external forcing term g belongs only to $L^{r}_{\mathrm {loc}}(\mathbb{R},L^{r}(\Omega ))$. Inspired by the idea for proving the existence of global attractors in $L^{r}(\Omega )$, we modify Theorem 5.11 [18] to prove the existence of the pullback attractors in $C_{\gamma ,L^{r}( \Omega )}$.

Lemma 5.1

Hypotheses (2), (3), (6) hold, and $g\in C(\mathbb{R};L^{2}(\Omega ))$. Then there exists a pullback attractor $\{\mathcal{A}_{C_{\gamma ,L^{2}(\Omega )}}(t)\}_{t\in \mathbb{R}}$ for the processes $\{U(t,\tau )\}$ on ${C_{\gamma ,L^{2}( \Omega )}}$ generated by the solution of Eq. (1).

Proof

By Theorem 13 [10], the processes $\{U(t,\tau )\}$ on $C_{\gamma ,H^{1}(\Omega )}$ associated with Eq. (1) has a pullback attractor $\mathcal{A}_{C_{\gamma ,H^{1}( \Omega )}}$. From the Sobolev embedding theorem $H^{1}(\Omega )\hookrightarrow \hookrightarrow L^{2}(\Omega )$ and $C_{\gamma ,H^{1}(\Omega )}\subseteq C_{\gamma ,L^{2}(\Omega )}$, $\mathcal{A}_{C_{\gamma ,H^{1}(\Omega )}}$ is a pullback attractor for the processes $\{U(t,\tau )\}$ on $C_{\gamma ,L^{2}(\Omega )}$. □

Lemma 5.2

Let $\{U(t,\tau )\}$ associated with Eq. (1) be an evolution process on $C_{\gamma ,L^{r}(\Omega )}$ with a pullback absorbing set $\mathcal{D}=\{D(t)\}_{t\in \mathbb{R}}$ on $C_{\gamma ,L^{r}(\Omega )}$. Then, for each $t\in \mathbb{R}$, for any $\varepsilon >0$, and any pullback absorbing set $\mathcal{D}\subset C_{\gamma ,L^{r}(\Omega )}$, there exist $T =T (\mathcal{D},t,\varepsilon )>0$, $M=M(\varepsilon )>0$ such that

$$\begin{aligned} m\bigl(\Omega^{\cdot }_{t}\bigl( \bigl\vert U(t,t+z)u^{0}(t+z) \bigr\vert \geq M\bigr)\bigr)\leq \varepsilon , \quad \textit{for any } -z\leq T , \textit{and } u^{0}_{t}(\cdot )\in \mathcal{D}, \end{aligned}$$

where $m(e)$ denotes the Lebesgue measure of $e\subset \Omega $ and $\Omega^{\cdot }_{t}( \vert u_{t}(z) \vert \geq M)\overset{\vartriangle }{=} \bigcup_{z\in (-\infty ,0]}\{x\in \Omega\mid \vert u(t+z,x) \vert \geq M\}$.

Proof

From the assumption that $\{U(t,\tau )\}$ has a pullback absorbing set in $C_{\gamma ,L^{r}(\Omega )}$, we know that there exists a positive constant $M_{0}$, such that, for each $t\in \mathbb{R}$ and for any pullback absorbing set $\mathcal{D}$ of $C_{\gamma ,L^{r}( \Omega )}$, we can find a positive constant T which depends on $\mathcal{D}$, such that

$$\begin{aligned} \bigl\Vert U(t,t+z)u^{0}(t+z) \bigr\Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \leq M_{0}, \quad \text{for any } -z\geq T , \text{and } u^{0}_{t}( \cdot )\in \mathcal{D}. \end{aligned}$$

So, we have

$$\begin{aligned} 2M_{0} \geq &2\sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega } \bigl\vert U(t,t+z)u ^{0}(t+z) \bigr\vert ^{r}\,dx \\ \geq & \sup_{z\in (-\infty ,-T_{1}]}e^{\gamma z} \int_{\Omega^{\cdot }_{t}(\{ \vert u(t+z) \vert \geq M_{1}\})} \bigl\vert U(t,t+z)u^{0}(t+z) \bigr\vert ^{r}\,dx \\ & {} +\sup_{z\in (-T_{1},0]}e^{\gamma z} \int_{\Omega^{\cdot }_{t}(\{ \vert u(t+z) \vert \geq M_{1}\})} \bigl\vert U(t,t+z)u^{0}(t+z) \bigr\vert ^{r}\,dx \\ \geq &e^{-\gamma T_{1}}\biggl( \int_{\Omega^{\cdot }_{t}(\{ \vert U(t,t+z)u^{0}(t+z) \vert \geq M_{1}\})}M_{1} ^{r}\,dx + \int_{\Omega^{\cdot }_{t}(\{ \vert U(t,t+z)u^{0}(t+z) \vert \geq M_{1}\})}M _{1}^{r}\,dx\biggr) \\ \geq & 2e^{-\gamma T_{1}}M_{1}^{r}m\bigl(\Omega \bigl(\bigl\{ \bigl\vert U(t,t+z)u^{0}(t+z) \bigr\vert \geq M_{1} \bigr\} \bigr)\bigr). \end{aligned}$$

This inequality implies that $m(\Omega^{\cdot }_{t}(\{ \vert U(t,t+z)u^{0}(t+z) \vert \geq M_{1}\}))\leq \varepsilon $, if we choose $M_{1}$ large enough such that $M_{1}\geq (\frac{M_{0}}{e^{-\gamma T_{1}}\varepsilon })^{ \frac{1}{r}}$. □

Lemma 5.3

For each $t\in \mathbb{R}$, any $\varepsilon >0$, the pullback absorbing set $\mathcal{D}$ of process $\{U(t,\tau )\}$ associated with Eq. (1) on $C_{\gamma ,L^{r}(\Omega )}$ ($r>0$) has a finite ε-net in $C_{\gamma ,L^{r}(\Omega )}$, if there exists a positive constant $M=M(\varepsilon )$ which depends on ε, such that

  1. (i)

    $\mathcal{D}$ has a finite $(3M)^{(2-r)/2}(\frac{\varepsilon }{2})^{ \frac{r}{2}}$-net in $C_{\gamma ,L^{2}(\Omega )}$,

  2. (ii)
    $$\begin{aligned} \begin{aligned}[b]& \biggl(\sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}(\{ \vert u(t+z) \vert \geq M\})} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx \biggr)^{ \frac{1}{r}}\\&\quad < 2^{-(2r+2)/r}\varepsilon , \quad \textit{for any } u_{t}(\cdot )\in \mathcal{D}.\end{aligned} \end{aligned}$$
    (61)

Proof

For each $t\in \mathbb{R}$, any fixed $\varepsilon >0$, it follows from the assumptions that $\mathcal{D}$ has a finite $\frac{(3M)^{(2-r)}}{2\varepsilon^{r/2}}$-net in $C_{\gamma ,L^{2}( \Omega )}$, that is, there exist $u_{t}^{1},\ldots,u_{t} ^{k}\in \mathcal{D}$, such that, for each $u_{t}(\cdot )\in \mathcal{D}$, we can find some $u_{t}^{i}$ ($1\leq i\leq k$) satisfying

$$\begin{aligned} \bigl\Vert u(t+z)-u^{i}(t+z) \bigr\Vert ^{2}_{L^{2}(\Omega )} \leq & \sup_{z\in (-\infty ,0]} e^{\gamma z} \bigl\Vert u(t+z)-u^{i}(t+z) \bigr\Vert ^{2}_{L ^{2}(\Omega )} \\ =&\sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u_{t}-u_{t}^{i} \bigr\Vert ^{2} _{L^{2}(\Omega )} < (3M)^{(2-r)} \biggl(\frac{\varepsilon }{2} \biggr)^{r}. \end{aligned}$$
(62)

Then, obviously, we have

$$\begin{aligned}& \bigl\Vert u_{t}-u_{t}^{i} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \\& \quad \leq \sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z)-u^{i}(t+z) \vert \geq 3M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad\quad {} +\sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z)-u^{i}(t+z) \vert \leq 3M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \end{aligned}$$
(63)

and

$$\begin{aligned}& \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z)-u^{i}(t+z) \vert \leq 3M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad \leq (3M)^{r-2}\sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u_{t}-u_{it} \vert \leq 3M)} \bigl\vert u_{t}-u_{t}^{i} \bigr\vert ^{2}\,dx, \\& \quad \leq (3M)^{r-2}(3M)^{2-r} \biggl(\frac{\varepsilon }{2} \biggr)^{r}= \biggl(\frac{\varepsilon }{2} \biggr)^{r}. \end{aligned}$$
(64)

On the other hand, set

$$\begin{aligned}& \Omega_{1}^{z}=\Omega^{z}_{t}\biggl( \bigl\vert u(t+z) \bigr\vert \geq \frac{3M}{2}\biggr)\cap \Omega _{t}^{z}\biggl( \bigl\vert u^{i}(t+z) \bigr\vert \leq \frac{3M}{2}\biggr), \\& \Omega_{2}^{z}=\Omega^{z}_{t}\biggl( \bigl\vert u(t+z) \bigr\vert \leq \frac{3M}{2}\biggr)\cap \Omega _{t}^{z}\biggl( \bigl\vert u^{i}(t+z) \bigr\vert \geq \frac{3M}{2}\biggr), \\ & \Omega_{3}^{z}=\Omega^{z}_{t} \biggl( \bigl\vert u(t+z) \bigr\vert \geq \frac{3M}{2}\biggr)\cap \Omega _{t}^{z}\biggl( \bigl\vert u^{i}(t+z) \bigr\vert \geq \frac{3M}{2}\biggr), \end{aligned}$$

then we have

$$\begin{aligned} \Omega^{z}_{t}\bigl( \bigl\vert u(t+z) \bigr\vert \geq 3M\bigr)\subset \Omega_{1}^{z}\cup \Omega_{2} ^{z}\cup \Omega_{3}^{z}. \end{aligned}$$

From the simple facts that $\vert u(t+z)-u^{i}(t+z) \vert \leq 2 \vert u(t+z) \vert $ in $\Omega_{1}^{z}$ and $\vert u(t+z)-u^{i}(t+z) \vert \leq 2 \vert u^{i}(t+z) \vert $ in $\Omega_{2}^{z}$, combining with (61), we have

$$\begin{aligned}& \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z)-u^{i}(t+z) \vert \geq 3M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\ & \quad \leq \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega_{1} ^{z}} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx+\sup_{z\in (-\infty ,0]} e^{ \gamma z} \int_{\Omega_{2}^{z}} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\ & \quad\quad {} + \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega_{3}^{z}} \bigl\vert u(t+z)-u ^{i}(t+z) \bigr\vert ^{r}\,dx \\ & \quad \leq 2^{r}\sup_{z\in (-\infty ,0]} e^{\gamma z} \biggl( \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx+ \int_{\Omega^{z}_{t}( \vert u^{i}(t+z) \vert \geq M)} \bigl\vert u^{i}(t+z) \bigr\vert ^{r}\,dx \\ & \quad\quad {} + \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx + \int_{\Omega_{t}^{z}( \vert u^{i}(t+z) \vert \geq M)} \bigl\vert u^{i}(t+z) \bigr\vert ^{r}\,dx \biggr) \\ & \quad \leq 2^{r+2}\cdot 2^{(2r+2)}\varepsilon^{r}= \biggl( \frac{\varepsilon }{2} \biggr)^{r}. \end{aligned}$$
(65)

Substituting (64) and (65) into (63), we can deduce that

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u(t+z)-u^{i}(t+z) \bigr\Vert _{L ^{r}(\Omega )} \leq \frac{\varepsilon }{2}+\frac{\varepsilon }{2}= \varepsilon , \end{aligned}$$

which means that $\mathcal{D}$ has a finite ε-net in $C_{\gamma ,L^{r}(\Omega )}$. □

Lemma 5.4

Let $\mathcal{D}$ be a pullback absorbing set in $C_{\gamma ,L^{r}( \Omega )}$ ($r\geq 1$). If $\mathcal{D}$ has a finite ε-net in $C_{\gamma ,L^{r}(\Omega )}$ ($r\geq 1$) then there exists a positive $M=M(B,\varepsilon )$, such that, for any $u_{t}(\cdot )\in \mathcal{D}$, $z\in (-\infty ,0]$, we can find

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx \leq 2^{r+1} \varepsilon^{r}. \end{aligned}$$

Proof

Since $\mathcal{D}$ has a finite ε-net in $C_{\gamma ,L^{r}(\Omega )}$ ($r\geq 1$), for each $t\in \mathbb{R}$, we know that there exist $u_{t}^{1},\ldots,u_{t}^{k}\in \mathcal{D}$, such that, for any $u_{t}(\cdot )\in \mathcal{D}$, we can find some $u_{t}^{i}$ ($1\leq i\leq k$) satisfying

$$\begin{aligned} \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx\leq \varepsilon^{r}. \end{aligned}$$
(66)

Simultaneously, for the fixed $\varepsilon >0$, there exists a $\delta >0$, such that, for each $u_{t}^{i}$, $1\leq i\leq k$, we have

$$\begin{aligned} \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{e} \bigl\vert u^{i}(t+z) \bigr\vert ^{r}\,dx \leq \varepsilon^{r}, \end{aligned}$$
(67)

provided that $m(e)<\delta$ ($e\subset \Omega $).

On the other hand, since $\mathcal{D}$ is bounded in $C_{\gamma ,L^{r}(\Omega )}$ ($r\geq 1$), for the fixed $\delta >0$ above, there exists $M>0$, such that $m(\Omega^{\cdot }_{t}( \vert u(t+z) \vert \geq M))< \delta $ holds for each $u_{t}\in B$. So, $m(\Omega^{z}_{t}( \vert u(t+z) \vert \geq M))<\delta $ also holds for each $u_{t}\in B$.

Therefore,

$$\begin{aligned}& \sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx \\& \quad = \sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z)-u^{i}(t+z)+u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad \leq 2^{r}\sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad\quad {} + 2^{r}\sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad \leq 2^{r+1}\varepsilon^{r}. \end{aligned}$$
(68)

 □

Lemma 5.5

For each $t\in \mathbb{R}$, for any $\varepsilon >0$ and any pullback absorbing set $\mathcal{D}\in C_{\gamma ,L^{2}(\Omega )}$, there exist two positive constants $T_{3}=T_{3}(B,\varepsilon )=\max \{T_{1},T _{2}\}$ and $M=M(\varepsilon )$, such that

$$\begin{aligned} \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z} _{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx< C \varepsilon , \quad \textit{for any } -z\geq T_{3},u^{0}_{t}( \cdot )\in \mathcal{D}, \end{aligned}$$
(69)

where the constant C is independent of ε and $\mathcal{D}$.

Proof

For each $t\in \mathbb{R}$, any fixed $\varepsilon >0$, there exists $\delta >0$ such that if $e\subset \Omega $ and $m(e)\leq \delta $, then

$$\begin{aligned} \int_{e} \bigl\vert \phi (x) \bigr\vert ^{r} \,dx \leq C\varepsilon , \end{aligned}$$
(70)

where $\phi (x),g(x)\in L^{r}{(\Omega )}$. Moreover, from Lemmas 5.1, 5.2 and 5.4, we know that there exist $T=T (\mathcal{D},\varepsilon )>0$ and $M=M(\varepsilon )$, for each $-z\geq T $, $u_{t}(\cdot )\in D$, we have

$$\begin{aligned} m\bigl(\Omega^{z}_{t}\bigl( \bigl\vert u(t+z) \bigr\vert \geq M\bigr)\bigr)< \min \{\varepsilon ,\delta \}, \quad \text{for each } t\in \mathbb{R}, \end{aligned}$$
(71)

and

$$\begin{aligned} \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert )\geq M} \bigl\vert u(t+z) \bigr\vert ^{2}< 8 \varepsilon . \end{aligned}$$
(72)

Thus, we also have

$$\begin{aligned} \int_{\Omega^{0}_{t}( \vert u(t) \vert \geq M)} \bigl\vert u(t) \bigr\vert ^{2}< 8 \varepsilon , \quad \text{for } t\in [T,+\infty ]. \end{aligned}$$
(73)

Multiplying (1) by $(u-M)_{+}^{r-1}$ and integrating over $\Omega^{0}_{t}=\Omega^{0}_{t}(u >M)$, we have

$$\begin{aligned}& \int_{\Omega^{0}_{t}(u >M)}\frac{\partial u}{\partial t}(u-M)_{+} ^{r-1} \,dx - \int_{\Omega^{0}_{t}(u >M)}\Delta u(u-M)_{+}^{r-1} \,dx \\& \quad\quad {} + \int_{\Omega^{0}_{t}(u >M)}\lambda u(u-M)_{+}^{r-1} \,dx \\& \quad = \int_{\Omega^{0}_{t}(u >M)}f(t,u_{t}) (u-M)_{+}^{r-1} \,dx + \int_{\Omega^{0}_{t}(u >M)}g(t,x) (u-M)_{+}^{r-1} \,dx. \end{aligned}$$
(74)

After integrating over $\Omega^{0}_{t}(u >M)$, (74) becomes

$$\begin{aligned}& \frac{1}{r}\frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} - \int_{\Omega^{0}_{t}(u >M)}\Delta u(u-M)_{+}^{r-1} \,dx + \lambda \int_{\Omega^{0}_{t}(u >M)}u(u-M)_{+}^{r-1} \,dx \\& \quad = \int_{\Omega^{0}_{t}(u >M)}F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) (u-M)_{+}^{r-1} \,dx+ \int_{\Omega^{0}_{t}( \vert u \vert >M)}g(t,x) (u-M)_{+}^{r-1} \,dx \\& \quad\quad {} + \int_{\Omega^{0}_{t}(u >M)} \int^{0}_{-\infty } \bigl\vert G\bigl(s,z,u(s+z)\bigr) \bigr\vert (u-M)_{+} ^{r-1}\,dz \,dx, \end{aligned}$$
(75)

where

$$ (u-M)_{+}= \textstyle\begin{cases} u-M,& u\geq M, \\ 0,& u\leq M. \end{cases} $$

Let $\Omega^{0}_{1,t}=\Omega^{0}_{t}(u >M)$, then we have

$$\begin{aligned}& \frac{1}{r}\frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} - \int_{ \Omega^{0}_{1,t}}\Delta u(u-M)_{+}^{r-1} \,dx+\lambda \int_{ \Omega^{0}_{1,t}}u(u-M)_{+}^{r-1} \,dx \\& \quad = \int_{ \Omega^{0}_{1,t}}F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) (u-M)_{+}^{r-1} \,dx+ \int_{ \Omega^{0}_{1,t}}g(t,x) (u-M)_{+}^{r-1} \,dx \\& \quad\quad {} + \int_{ \Omega^{0}_{1,t}} \int^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr) (u-M)_{+} ^{r-1} \,dz \,dx. \end{aligned}$$

We now estimate every term of (75). First, we obtain

$$\begin{aligned} - \int_{ \Omega^{0}_{1,t}}\Delta u(u-M)_{+}^{r-1} \,dx=(r-1) \int_{\Omega^{0}_{1}}\nabla u \bigl\vert (u-M)_{+} \bigr\vert ^{r-2}\nabla u \,dx\geq 0 \end{aligned}$$
(76)

and

$$\begin{aligned} \lambda \int_{ \Omega^{0}_{1,t}}u(u-M)_{+}^{r-1}\,dx\geq \lambda \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )}. \end{aligned}$$
(77)

By the assumption (2), (3), (6) and Young’s inequality, we have

$$\begin{aligned}& \int_{ \Omega^{0}_{1,t}}F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) (u-M)_{+}^{r-1}\,dx \\ & \quad \leq \frac{\varepsilon^{-(r-1)}_{1}}{r} \int_{ \Omega^{0}_{1,t}} \bigl\vert F\bigl(x,u\bigl(x,t- \rho (t)\bigr) \bigr) \bigr\vert ^{r}\,dx +\frac{(r-1)\varepsilon_{1}}{r} \int_{ \Omega^{0} _{1,t}}(u-M)_{+}^{r}\,dx \\ & \quad \leq \frac{ \varepsilon^{-(r-1)}_{1}}{r} \int_{ \Omega^{0}_{1,t}} \vert k _{1} \vert ^{r}\,dx + \frac{ k^{r}_{2}\varepsilon^{-(r-1)}_{1}}{r} \int_{ \Omega^{0}_{1,t}}e^{-r\gamma \rho (t)} \bigl\vert u\bigl(x,t-\rho (t) \bigr) \bigr\vert ^{r}\,dx \\ & \quad\quad {} +\frac{(r-1)\varepsilon_{1}}{r} \int_{ \Omega^{0}_{1,t}}(u-M)_{+} ^{r}\,dx \\ & \quad \leq \frac{ \varepsilon^{-(r-1)}_{1}}{r} \vert k_{1} \vert ^{r}| _{L^{r}( \Omega ^{0}_{1,t})} +\frac{ k^{r}_{2}\varepsilon^{-(r-1)}_{1}}{r} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1})}}+ \frac{(r-1)\varepsilon_{1}}{r} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}( \Omega^{0} _{1,t})} , \end{aligned}$$
(78)
$$\begin{aligned}& \int_{ \Omega^{0}_{1,t}} \int^{0}_{-\infty }G\bigl(x,z,u(s+z)\bigr) (u-M)_{+} ^{r-1} \,dz \,dx \\ & \quad \leq \int_{ \Omega^{0}_{1,t}} \int^{0}_{-\infty } \bigl\vert m_{0}(z) \bigr\vert \bigl\vert (u-M)_{+} \bigr\vert ^{r-1}\,dz \,dx + \int_{ \Omega^{0}_{1,t}} \int^{0}_{-\infty }m_{1}(z)| u(t+z ) (u-M)_{+} ^{r-1}\,dz\,dx \\ & \quad \leq \frac{ \varepsilon^{-(r-1)}_{2}}{r} \int_{ \Omega^{0}_{1,t}} \vert m _{0} \vert ^{r}\,dx + \frac{(r-1)\varepsilon_{2}}{r} \int_{\Omega_{1}}(u-M)_{+} ^{r}\,dx \\ & \quad\quad {} +\frac{\bar{m}_{1}\varepsilon^{-(r-1)}_{3}}{r} \int_{ \Omega^{0}_{1,t}} \bigl\vert u(t+z ) \bigr\vert ^{r}\,dx + \frac{\bar{m}_{1}(r-1)\varepsilon _{3}}{r} \int_{ \Omega^{0}_{1,t}}(u-M)_{+}^{r}\,dx \\ & \quad \leq \frac{ \varepsilon^{-(r-1)}_{2}}{r} \vert m_{0} \vert ^{r}_{L^{r}( \Omega ^{0}_{1,t})}+ \frac{(r-1)\varepsilon_{2}}{r} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})} \\ & \quad\quad {} +\frac{\bar{m}_{1}(r-1)\varepsilon_{3}}{r} \bigl\Vert u(t+z ) \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}, \end{aligned}$$
(79)

and

$$\begin{aligned} \int_{ \Omega^{0}_{1,t}}g(t,x) (u-M)_{+}^{r-1} \,dx &\leq \int_{\Omega^{0}_{1,t}} \bigl\vert g(t,x) \bigr\vert (u-M)_{+}^{r-1}\,dx \\ & \leq \frac{\varepsilon^{-(r-1)}_{4}}{r} \int_{\Omega^{0}_{1,t}} \bigl\vert g(t,x) \bigr\vert ^{r}\,dx + \frac{(r-1)\varepsilon_{4}}{r} \int_{\Omega^{0}_{1,t}}(u-M)_{+}^{r}\,dx \\ & \leq \frac{\varepsilon^{-(r-1)}_{4}}{r} \bigl\Vert g(t,x) \bigr\Vert ^{r}_{L^{r}( \Omega ^{0}_{1,t})} +\frac{(r-1)\varepsilon_{4}}{r} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}. \end{aligned}$$
(80)

Combining with (76)–(80), we can conclude that

$$\begin{aligned}& \frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} +r(r-1) \int_{ \Omega^{0}_{1,t}}\nabla u(u-M)_{+}^{r-2}\nabla u \,dx \\ & \quad\quad {} +r\lambda \int_{ \Omega^{0}_{1,t}}u(u-M)_{+}^{r-1}\,dx \\ & \quad \leq \varepsilon^{-(r-1)}_{1} \int_{ \Omega^{0}_{1,t}} \vert k_{1} \vert ^{r}\,dx + \varepsilon^{-(r-1)}_{2} \int_{ \Omega^{0}_{1,t}} \vert m_{0} \vert ^{r}\,dx \\ & \quad\quad {} +(r-1) (\varepsilon_{1}+\varepsilon_{2}+ {m}_{1}\varepsilon_{3}+ \varepsilon_{4}) \int_{ \Omega^{0}_{1,t}}(u-M)_{+}^{r}\,dx \\ & \quad\quad {} + k^{r}_{2}\varepsilon^{-(r-1)}_{1}e^{-r\gamma \rho (t)} \int_{ \Omega^{0}_{1,t}} \bigl\vert u\bigl(x,t-\rho (t)\bigr) \bigr\vert ^{r}\,dx \\ & \quad\quad {} + {m}_{1}\varepsilon^{-(r-1)}_{3} \int_{ \Omega^{0}_{1,t}}e^{\gamma z} \bigl\vert u(t+z ) \bigr\vert ^{r}\,dx +\varepsilon^{-(r-1)}_{4} \int_{ \Omega^{0}_{1,t}} \bigl\vert g(t,x) \bigr\vert ^{r}\,dx. \end{aligned}$$
(81)

We also have

$$\begin{aligned}& \frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq -r\lambda \bigl\Vert (u-M)_{+} \bigr\Vert _{r}^{r}+ \varepsilon^{-(r-1)}_{1} \vert k _{1} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}+\varepsilon^{-(r-1)}_{2} \vert m_{0} \vert ^{r} _{L^{r}( \Omega^{0}_{1,t})} \\& \quad\quad {} +(r-1) (\varepsilon_{1}+\varepsilon_{2}+ {m}_{1}\varepsilon_{3}+ \varepsilon_{4}) \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}+ k^{r} _{2}\varepsilon^{-(r-1)}_{1} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega ^{0}_{1,t})}} \\& \quad\quad {} + {m}_{1}\varepsilon^{-(r-1)}_{3} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{0}_{1,t})}}+\varepsilon^{-(r-1)}_{4} \bigl\Vert g(t,x) \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}. \end{aligned}$$
(82)

Let $\alpha >0$, which will also be determined later. Then

$$\begin{aligned}& \frac{d}{dt}e^{\alpha t} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad =\alpha e^{\alpha t} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{r} +e^{\alpha t}\frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{r} \\& \quad \leq -\bigl(r\lambda -\alpha -(r-1) (\varepsilon_{1}+ \varepsilon_{2}+ {m}_{1}\varepsilon_{3}+ \varepsilon_{4})\bigr) e^{\alpha t} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad\quad {} +\bigl( \varepsilon^{-(r-1)}_{1} \vert k_{1} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}+ \varepsilon^{-(r-1)}_{2} \vert m_{0} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}\bigr) e^{ \alpha t} +\varepsilon^{-(r-1)}_{4}e^{\alpha t} \bigl\Vert g(t,x) \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})} \\& \quad\quad {} +\bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1}\varepsilon^{-(r-1)} _{3}\bigr)e^{\alpha t} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{0}_{1,t})}}. \end{aligned}$$
(83)

Let $A=(r\lambda -\alpha -(r-1)(\varepsilon_{1}+\varepsilon_{2}+ {m}_{1}\varepsilon_{3}+\varepsilon_{4}))$. By Gronwall’s inequality, we have

$$\begin{aligned}& e^{\alpha t} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega_{t,0}^{1} )} \\& \quad \leq e^{-A(t-\tau )}e^{\alpha \tau } \bigl\Vert \bigl(u(\tau )-M \bigr)_{+} \bigr\Vert ^{r}_{C _{\gamma ,L^{r}(\Omega )}} + \varepsilon^{-(r-1)}_{4}e^{-At} \int^{t} _{-\infty }e^{(A+\alpha )s} \bigl\Vert g(s,x) \bigr\Vert ^{r}_{L^{r}(\Omega^{0}_{1,t})}\,ds \\& \quad\quad {} +\bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1}\varepsilon^{-(r-1)} _{3}\bigr)e^{-At} \int^{t}_{\tau }e^{(A+\alpha )s} \Vert u_{s} \Vert ^{r}_{C_{ \gamma ,L^{r}(\Omega^{0}_{1,t})}}\,ds \\& \quad\quad {} +\bigl( \varepsilon^{-(r-1)}_{1} \vert k_{1} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}+ \varepsilon^{-(r-1)}_{2} \vert m_{0} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}\bigr) \frac{e ^{\alpha t}}{A+\alpha }. \end{aligned}$$
(84)

Thanks to (46), and letting $\alpha_{1}>\alpha \geq \alpha ^{*}$, we can deduce that

$$\begin{aligned}& \bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1}\varepsilon^{-(r-1)} _{3} \bigr)e^{-At} \int^{t}_{\tau } e^{(A+\alpha )s} \Vert u_{s} \Vert ^{r}_{C _{\gamma ,L^{r}(\Omega_{t,0}^{1} )}}\,ds \\& \quad \leq \bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1} \varepsilon^{-(r-1)}_{3} \bigr) \biggl( \frac{re^{\alpha \tau }}{A} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega_{t,0}^{1})}} + \frac{\alpha C_{ \Omega_{t,0}^{1}}e^{\alpha t}}{(A+\alpha )(\alpha -L)} \\& \quad\quad {} + \varepsilon_{4}^{-(r-1)}\frac{1}{A} \int^{t}_{-\infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega_{t,0}^{1})}^{r}\,ds +\frac{r e^{(\alpha -L) \tau }e^{Lt}}{(A+L)} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega_{t,0}^{1})}} \\& \quad\quad {} +\varepsilon_{4}^{-(r-1)}\frac{e^{Lt}}{(A+L)} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega_{t,0}^{1})}^{r} \bigr)\,ds \biggr). \end{aligned}$$
(85)

Multiplying (84) by $e^{-\alpha t}$, we have

$$\begin{aligned}& \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega_{t,0}^{1} )} \\& \quad \leq e^{-A(t-\tau )}e^{\alpha \tau }e^{-\alpha t} \bigl\Vert \bigl(u( \tau )-M\bigr)_{+} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}}+ \frac{\varepsilon^{-(r-1)} _{1} \vert k_{1} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}}{A+\alpha } \\& \quad\quad {} +\varepsilon^{-(r-1)}_{4}e^{-(A+\alpha )t} \int^{t}_{-\infty }e^{(A+ \alpha )s} \bigl\Vert g(s,x) \bigr\Vert ^{r}_{L^{r}(\Omega^{0}_{1,t})}\,ds+\frac{ \varepsilon^{-(r-1)}_{2} \vert m_{0} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}}{A+ \alpha } \\& \quad\quad {} + \bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1}\varepsilon^{-(r-1)} _{3} \bigr) \biggl( \frac{re^{\alpha \tau }e^{-\alpha t}}{A} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}} + \frac{\alpha C_{\Omega ^{0}_{1,t}}}{(A+\alpha )(\alpha -L)} \\& \quad\quad {} + \varepsilon_{4}^{-(r-1)}\frac{1}{A} e^{-\alpha t} \int^{t}_{- \infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega^{0}_{1,t})}^{r}\,ds +\frac{r e^{(\alpha -L)\tau }e^{-(\alpha -L)t}}{(A+L)} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}} \\& \quad\quad {} +\varepsilon_{4}^{-(r-1)}\frac{e^{-(\alpha -L)t}}{(A+L)} \int^{t} _{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega^{0}_{1,t})} ^{r} \bigr)\,ds \biggr) \\& \quad \leq e^{\alpha \tau }e^{-\alpha t} \bigl\Vert (\phi -M)_{+} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t} )}} +Ce^{-(A+\alpha )t} \int^{t}_{-\infty }e ^{(A+\alpha )s} \bigl\Vert g(s,x) \bigr\Vert ^{r}_{L^{r}(\Omega^{0}_{1,t})}\,ds \\& \quad\quad {} + C m\bigl({\Omega^{0}_{1,t}}\bigr) +Ce^{\alpha \tau }e^{-\alpha t} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}} +CC_{\Omega^{0}_{1,t}} \\& \quad\quad {} + C e^{-\alpha t} \int^{t}_{-\infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega^{0}_{1,t})}^{r}\,ds +Ce^{(\alpha -L)\tau }e^{-(\alpha -L)t} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}} \\& \quad\quad {} +Ce^{-(\alpha -L)t} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega^{0}_{1,t})}^{r} \bigr)\,ds. \end{aligned}$$
(86)

Now replacing t by $t+z$, similar to the arguments in Lemma 4.1, in view of (45), we have

$$\begin{aligned}& e^{r\gamma z} \bigl\Vert (u_{t}-M)_{+} \bigr\Vert ^{r}_{{L^{r}( \Omega^{z}_{1,t} )}} \\& \quad \leq e^{\alpha \tau }e^{-\alpha t}e^{(r\gamma -\alpha )z} \bigl\Vert ( \phi -M)_{+} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{z}_{1,t} )}} +Ce^{-(A+ \alpha )t} \int^{t}_{-\infty }e^{(A+\alpha )s} \bigl\Vert g(s,x) \bigr\Vert ^{r}_{L^{r}( \Omega^{z}_{1,t})}\,ds \\& \quad\quad {} + C m\bigl({ \Omega^{z}_{1,t}}\bigr)e^{(r\gamma -\alpha )z} +Ce^{\alpha \tau }e^{-\alpha t} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{z}_{1,t})}} +CC_{ \Omega^{z}_{1,t}} \\& \quad\quad {} + C e^{-\alpha t}e^{(r\gamma -\alpha )z} \int^{t}_{-\infty }e^{ \alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega^{z}_{1,t})}^{r}\,ds +Ce^{(\alpha -L) \tau }e^{-(\alpha -L)t}e^{(r\gamma +L-\alpha )z} \Vert \phi \Vert ^{r}_{C_{ \gamma ,L^{r}( \Omega^{z}_{1,t})}} \\& \quad\quad {} +Ce^{-(\alpha -L)t}e^{(r\gamma +L-\alpha )z} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega^{z}_{1,t})}^{r} \bigr)\,ds. \end{aligned}$$
(87)

Furthermore, by (57) and (70), we have

$$\begin{aligned}& \bigl\Vert (u_{t}-M)_{+} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{z}_{1,t} )}} \\& \quad \leq e^{\alpha \tau }e^{-\alpha t}\varepsilon +C\varepsilon e^{-(A+ \alpha )t} \int^{t}_{-\infty }e^{(A+\alpha )s} \,ds+ C\varepsilon +Ce ^{\alpha \tau }e^{-\alpha t}\varepsilon +C\varepsilon \\& \quad\quad {} + C e^{-\alpha t}\varepsilon \int^{t}_{-\infty }e^{\alpha s} \,ds +Ce ^{(\alpha -L)\tau }e^{-(\alpha -L)t}\varepsilon +Ce^{-(\alpha -L)t} \varepsilon \int^{t}_{-\infty } e^{(\alpha -L)s}\,ds \\& \quad \leq e^{\alpha \tau }e^{-\alpha t}\varepsilon +C\varepsilon + C \varepsilon +Ce^{\alpha \tau }e^{-\alpha t}\varepsilon +C\varepsilon + C \varepsilon +Ce^{(\alpha -L)\tau }e^{-(\alpha -L)t}\varepsilon +C \varepsilon \\& \quad \leq C\varepsilon, \end{aligned}$$
(88)

where $\alpha >L$. Repeating the same steps above, just taking $(u(t+z )-M)_{-}$ instead of $(u(t+z )-M)_{+}$, we deduce that

$$\begin{aligned} \bigl\Vert \bigl(u(t+z )-M\bigr)_{-} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{z}_{1,t} )}}\leq C \varepsilon . \end{aligned}$$
(89)

From (88), (89) and Lemma 5.1, we know the hypotheses of Lemma 5.3 are all satisfied. Therefore the process $\{U(t,\tau )\}$ generated by Eq. (1) is $\mathcal{D}$-pullback ω-limit compact. □

Theorem 5.6

Suppose in addition to the hypotheses in Lemma 4.1 that $g\in C(\mathbb{R},L^{r}(\Omega ))$. Then the processes $\{U(t,\tau ) \}$ on $C_{\gamma , L^{r}(\Omega )}$ generated by the solution of Eq. (1) with $u_{0}\in C_{\gamma , L^{r}(\Omega )}$ has the $\mathcal{D}$-pullback attractors $\{\mathcal{A}_{C_{\gamma , L^{r}( \Omega )}}(t)\}_{t\in \mathbb{R}}$.

Proof

From Theorem 7.1, Lemmas 4.1, 5.1 and 5.5, now for every bounded subset B in $C_{\gamma ,L^{r}(\Omega )}$, the process generated by Eq. (1) has the pullback attractors in $C_{\gamma ,L^{r}(\Omega )}$. □

Uniform estimates in $C_{\gamma,W^{1,r}(\Omega)}$

Let semigroup $e^{At}$ has the following higher smooth effect [19]:

$$\begin{aligned} \bigl\Vert e^{At}x \bigr\Vert _{E^{\beta }_{r}}\leq Mt^{-(\beta -\alpha )}e^{-\delta t} \Vert x \Vert _{E^{\alpha }_{r}},\quad x\in E^{\beta }_{r}, t>0, 0\leq \alpha \leq \beta , 0< \delta < \lambda_{1}. \end{aligned}$$
(90)

Lemma 6.1

Suppose the conditions of Lemma 4.1 hold and

$$\begin{aligned} \alpha < r(\delta -\eta )\leq r\gamma ,\quad r>2, \end{aligned}$$
(91)

holds, the family of processes $\{U_{g}(t,\tau )\}$ is uniformly dissipative in $C_{\gamma ,W^{1,r}(\Omega )}$, where $g(x,t)\in L^{r} _{\mathrm {loc}}{(\mathbb{R};L^{r}(\Omega ))}$, $\eta >0$ will be determined later.

Proof

Choosing $\alpha_{1}$ with $\alpha <\alpha_{1}$ and using (46), we obtain

$$\begin{aligned}& \int^{t}_{\tau }e^{-\alpha_{1}(t-s)} \Vert u_{s} \Vert _{C_{L^{r}(\Omega )}} ^{r}\,ds \\& \quad \leq \int^{t}_{\tau }e^{-\alpha_{1}(t-s)} \biggl(r e^{\alpha \tau }e^{-\alpha s} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+ \frac{ \alpha }{\alpha -L}C_{\Omega } \\& \quad\quad {} + \varepsilon_{4}^{-(r-1)}e^{-\alpha s} \int^{s}_{-\infty }e^{ \alpha l} \bigl\Vert g(l) \bigr\Vert _{L^{r}(\Omega )}^{r}\,dl +r e^{(\alpha -L)\tau }e^{(L- \alpha )s} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}} \\& \quad\quad {} + \varepsilon_{4}^{-(r-1)}e^{(L-\alpha )s} \int^{s}_{-\infty } e^{( \alpha -L)l} \bigl\Vert g(l) \bigr\Vert _{L^{r}(\Omega )}^{r}\,dl \biggr)\,ds \\& \quad \leq \frac{C}{\alpha_{1}-\alpha } e^{\alpha \tau } \Vert \phi \Vert ^{r} _{C_{\gamma ,{L^{r}(\Omega )}}}+ C+ \frac{C}{\alpha_{1}-\alpha }e^{- \alpha t} \int^{t}_{-\infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )} ^{r}\,ds \\& \quad\quad {} + \frac{C e^{(\alpha -L)\tau }e^{(L-\alpha )t}}{\alpha_{1}-\alpha +L} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}} + \frac{C e^{(L-\alpha )t}}{ \alpha_{1}-\alpha +L} \int^{t}_{-\infty } e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L ^{r}(\Omega )}^{r}\,ds \\& \quad \overset{\vartriangle }{=} Q(\alpha_{1},\alpha ,L,\tau ,\phi ,g_{0},t). \end{aligned}$$
(92)

It is obvious that $Q(\alpha_{1},\alpha ,L,\tau ,\phi ,g_{0},t)$ is bounded, as $\tau \rightarrow -\infty $. From the well-posedness of (1), we know that the solution of (1) satisfies

$$\begin{aligned} u(t)=e^{A(t-\tau )}u(\tau )+ \int^{t}_{\tau }e^{A(t-s)}\bigl[-\lambda u+f(x,u _{s})+g(x,s)\bigr]\,ds. \end{aligned}$$
(93)

Therefore, using (90) and choosing $\alpha_{1}>0$, $\eta >0$, $q=\frac{r}{r-1}<2$, $r>2$ such that $0<\alpha <r(\delta -\eta )=\alpha_{1}<r\gamma $, for each $t\geq \tau $ we obtain

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{W^{1,r}{(\Omega )}}&= \biggl\Vert e^{A(t-\tau )}u(\tau )+ \int^{t}_{ \tau }e^{A(t-\tau )}\bigl[-\lambda u+f(x,u_{s})+g(x,s)\bigr]\,ds \biggr\Vert _{W^{1,r}{( \Omega )}} \\ & \leq \bigl\Vert e^{A(t -\tau )} u(\tau ) \bigr\Vert _{W^{1,r}{(\Omega )}}+ \lambda \int^{t }_{\tau } \bigl\Vert e^{A(t -s)} u \bigr\Vert _{W^{1,r}{(\Omega )}}\,ds \\ &\quad {} + \int^{t}_{\tau } \bigl\Vert e^{A(t-s)}f(x,u_{s}) \bigr\Vert _{W^{1,r}{(\Omega )}}\,ds + \int^{t}_{\tau } \bigl\Vert e^{A(t+z-s)}g(x,s) \bigr\Vert _{W^{1,r}{(\Omega )}}\,ds \\ & \leq M_{1}e^{-\delta (t-\tau )} \bigl\Vert u(\tau ) \bigr\Vert _{W^{1,r}{(\Omega )}}+ \lambda M_{2} \int^{t}_{\tau }(t-s)^{-\frac{1}{2}}e^{-\delta (t-s)} \Vert u \Vert _{L^{r}{(\Omega })}\,ds \\ &\quad {} +M_{3} \int^{t}_{\tau }(t-s)^{-\frac{1}{2}}e^{-\delta (t-s)} \bigl\Vert F\bigl(s,u\bigl(s- \rho (s)\bigr)\bigr) \bigr\Vert _{L^{r}{(\Omega })}\,ds \\ &\quad {} +M_{4} \int^{t}_{\tau }(t-s)^{-\frac{1}{2}}e^{-\delta (t-s)} \biggl\Vert \int ^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr)\,dz \biggr\Vert _{L^{r}{(\Omega })}\,ds \\ &\quad {} +M_{5} \int^{t }_{\tau }(t-s)^{-\frac{1}{2}}e^{-\delta (t-s)} \bigl\Vert g(x,s) \bigr\Vert _{L^{r}{(\Omega })}\,ds. \end{aligned}$$
(94)

Then, by (46), (92), Hold’s inequality and Young’s inequality, we have

$$\begin{aligned}& \lambda M_{2} \int^{t}_{\tau }(t -s)^{-\frac{1}{2}}e^{-\delta (t -s)} \Vert u \Vert _{L^{r}{(\Omega })}\,ds \\& \quad \leq \lambda M_{2} \biggl( \int^{t}_{\tau }(t -s)^{-\frac{1}{2}q}e ^{-q\eta (t -s)}\,ds \biggr)^{\frac{1}{q}} \times \biggl( \int^{t}_{\tau }e ^{-r(\delta -\eta )(t -s)} \Vert u \Vert ^{r}_{L^{r}{(\Omega })} \biggr)^{ \frac{1}{r}} \\& \quad \leq \frac{\lambda M_{2}}{q} \biggl( \int^{t }_{\tau }(t -s)^{- \frac{1}{2}q}e^{-q\eta (t -s)}\,ds \biggr)+\frac{\lambda M_{2}}{r} \biggl( \int^{t}_{\tau }e^{-r(\delta -\eta )(t-s)} \Vert u \Vert ^{r}_{L^{r}{(\Omega })}\,ds \biggr) \\& \quad \leq \frac{\lambda M_{2}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}}+\frac{\lambda M_{2}}{r} \biggl( \int^{t}_{\tau }e ^{-r(\delta -\eta )(t-s)} \Vert u \Vert ^{r}_{L^{r}{(\Omega })}\,ds \biggr) \\& \quad \leq \frac{\lambda M_{2}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}}+\frac{\lambda M_{2}}{r}Q\bigl(r(\delta -\eta ), \tau ,\phi ,g_{0},t\bigr) \\& \quad \overset{\vartriangle }{=} \frac{\lambda M_{2}\Gamma (1-\frac{q}{2})}{q ^{2-\frac{1}{2}q}\eta^{1-\frac{1}{2}q}} +R_{2,W^{1,r}(\Omega )}\bigl(r( \delta -\eta ),\tau ,\phi ,g_{0},t\bigr). \end{aligned}$$
(95)

Similarly, combining (2), (3), and (6), we have

$$\begin{aligned}& M_{3} \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e^{-\delta (t -s)} \bigl\Vert F\bigl(x,u\bigl(s- \rho (s)\bigr)\bigr) \bigr\Vert _{L^{r}{(\Omega })}\,ds \\& \quad \leq M_{3} \biggl( \int^{t }_{\tau }(t -s)^{-\frac{1}{2}q}e^{-q \eta (t -s)}\,ds \biggr)^{\frac{1}{q}} \times \biggl( \int^{t}_{\tau }e^{-r( \delta -\eta )(t -s)} \Vert F \Vert ^{r}_{L^{r}{(\Omega })} \biggr)^{\frac{1}{r}} \\& \quad \leq \frac{ M_{3}}{q} \biggl( \int^{t }_{\tau }(t -s)^{-\frac{1}{2}q}e ^{-q\eta (t -s)}\,ds \biggr)+\frac{ M_{3}}{r} \biggl( \int^{t }_{\tau }e^{-r( \delta -\eta )(t -s)} \Vert F \Vert ^{r}_{L^{r}{(\Omega })}\,ds \biggr) \\& \quad \leq \frac{M_{3}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}} +\frac{ M_{3}}{r} \int^{t}_{\tau }e^{-r(\delta -\eta )(t-s)} \bigl(k_{1}^{r} \Vert \Omega \Vert ^{r}_{L^{r}{(\Omega })}+k_{2} ^{r} e^{-r\gamma \rho (t)} \bigl\Vert u\bigl(s-\rho (s)\bigr) \bigr\Vert ^{r}_{L^{r}{(\Omega })} \bigr)\,ds \\& \quad \leq \frac{M_{3}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}} +\frac{M_{3} k_{1}^{r} \vert \Omega \vert ^{r}_{L^{r}( \Omega )}}{r^{2}(\delta -\eta )} +\frac{k_{2}^{r}M_{3}}{r}Q\bigl(r(\delta -\eta ),\tau ,\phi ,g_{0},h,t\bigr) \\& \quad \overset{\vartriangle }{=} \frac{ M_{3}\Gamma (1-\frac{r}{2})}{q^{2- \frac{1}{2}q}\eta^{1-\frac{1}{2}q}} +\frac{M_{3} k_{1}^{r} \vert \Omega \vert ^{r} _{L^{r}(\Omega )}}{r^{2}(\delta -\eta )} +R_{3,W^{1,r}(\Omega )}\bigl(r( \delta -\eta ),\tau ,\phi ,g_{0},t\bigr), \end{aligned}$$
(96)
$$\begin{aligned}& M_{4} \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e^{-\delta (t -s)} \biggl\Vert \int^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr)\,dz \biggr\Vert _{L^{r}{(\Omega })}\,ds \\& \quad \leq M_{4} \biggl( \int^{t }_{\tau }(t-s)^{-\frac{1}{2}q}e^{-q\eta (t -s)}\,ds \biggr)^{\frac{1}{q}} \\& \quad\quad {} \times \biggl( \int^{t }_{\tau }e^{-r(\delta -\eta )(t -s)} \biggl\Vert \int ^{0}_{-\infty }\bigl(m_{0}(z)+{m}_{1}(z) \bigl\vert u(s+z_{0}) \bigr\vert \bigr)\,dz \biggr\Vert _{L^{r}{(\Omega })} ^{r} \biggr)^{\frac{1}{r}} \\& \quad \leq \frac{M_{4}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}}+\frac{ M_{4}}{r} \biggl(m_{0}^{r} \vert \Omega \vert ^{r} \int^{t}_{\tau }e^{-r(\delta -\eta )(t-s)} \,ds+{m}_{1}^{r} \int^{t}_{ \tau }e^{-r(\delta -\eta )(t-s)} \Vert u_{s} \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega )}}\,ds \biggr) \\& \quad \overset{\vartriangle }{=} \frac{M_{4}\Gamma (1-\frac{q}{2})}{q^{2- \frac{1}{2}q}\eta^{1-\frac{1}{2}q}}+\frac{2^{r-1} M_{4} m_{0}^{r} \vert \Omega \vert ^{r}}{r^{2}(\delta -\eta )}+R_{4,W^{1,r}(\Omega )} \bigl(r(\delta - \eta ), \tau ,\phi ,g_{0},t\bigr), \end{aligned}$$
(97)

and

$$\begin{aligned}& \int^{t }_{\tau } \bigl\Vert e^{A(t -s)} g(x,s) \bigr\Vert _{W^{1,r}{(\Omega )}}\,ds \\& \quad \leq M_{5} \int^{t}_{\tau }(t -s)^{-\frac{1}{2}}e^{-\delta (t -s)} \Vert g \Vert _{L^{r} (\Omega )}\,ds \\& \quad \leq M_{5} \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e^{-(\delta - \eta )(t -s)}e^{-\delta (t -s)} \Vert g \Vert _{L^{r}(\Omega )}\,ds \\& \quad \leq M_{5} \biggl( \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e^{-q\delta (t -s)}\,ds \biggr)^{\frac{1}{q}}\times \biggl( \int^{t}_{\tau }e^{-r( \delta -\eta )(t -s)} \Vert g \Vert ^{r}_{L^{r}(\Omega )}\,ds \biggr)^{\frac{1}{r}} \\& \quad \leq \frac{M_{5}}{q} \biggl( \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e ^{-q\delta (t -s)}\,ds \biggr)+ \frac{M_{5}}{r} \biggl( \int^{t}_{-\infty }e ^{-r(\delta -\eta )(t -s)} \Vert g \Vert ^{r}_{L^{r}(\Omega )}\,ds \biggr) \\& \quad \overset{\vartriangle }{=} \frac{ M_{5}\Gamma (1-\frac{q}{2})}{q^{2- \frac{1}{2}q}\eta^{1-\frac{1}{2}q}} + R_{5,W^{1,r}(\Omega )}\bigl(r( \delta -\eta ),\tau ,q,g,t\bigr). \end{aligned}$$
(98)

Similar to the arguments in Lemma 4.1, for each $t\in \mathbb{R}$, we can conclude that by (91)

$$\begin{aligned}& \sup_{z\in [-\infty ,0]}e^{-r\gamma z} \bigl\Vert u(t+z) \bigr\Vert _{W^{1,r}{(\Omega )}} \\& \quad \leq M_{1}e^{-\delta (t-\tau )} \bigl\Vert u(\tau ) \bigr\Vert _{W^{1,r}{(\Omega )}}+\frac{( \lambda M_{2}+M_{3}+M_{4}+M_{5})\Gamma (1-\frac{r}{2})}{r^{2- \frac{1}{2}r} \eta^{1-\frac{1}{2}r}} \\& \quad\quad {} +R_{2,W^{1,r}{(\Omega )}}\bigl(r(\delta -\eta ),\tau ,\phi ,g_{0} ,t\bigr) +\frac{M _{3} k_{1}^{r} \vert \Omega \vert ^{r}}{r^{2}(\delta -\eta )} \\& \quad\quad {} +R_{3,W^{1,r}}\bigl(r(\delta -\eta ),\phi ,\tau ,g_{0} ,t \bigr)+\frac{2^{r-1} M_{4} m_{0}^{r} \vert \Omega \vert ^{r}}{r^{2}(\delta -\eta )} \\& \quad\quad {} +R_{4,W^{1,r}}\bigl(r(\delta -\eta ),\tau ,\phi ,g_{0} ,t \bigr)+ R_{5,W ^{1,r}(\Omega )}\bigl(r(\delta -\eta ),\tau ,q,g,t\bigr) \\& \quad \overset{\vartriangle }{=} R_{6,W^{1,r}(\Omega )}\bigl(r(\delta -\eta ), \tau ,r, \phi ,g_{0}, t\bigr),\quad \text{for each } t\in \mathbb{R}. \end{aligned}$$
(99)

Hence, we can see that $\sup_{z\in [-\infty ,0]}e^{-r\gamma z} \Vert u(t+z) \Vert _{W^{1,r}{(\Omega )}}$ is bounded, for each $t\in \mathbb{R}$, $z\in (-\infty ,0]$, as $\tau \rightarrow -\infty $, which implies the process $\{U(t,\tau )\}$ has pullback absorbing sets in $C_{\gamma ,W^{1,r}(\Omega )}$. □

Existence of the pullback attractors in $C_{\gamma,W^{1,r}(\Omega)}$

Theorem 7.1

Suppose in additional to the hypotheses in Lemma 6.1 and $g(s)\in C( \mathbb{R}, W^{1,r}(\Omega ))$, $F\in C^{1}(\mathbb{R}\times \mathbb{R}; \mathbb{R})$, $G\in C^{1}(\mathbb{R} \times \mathbb{R}\times \mathbb{R};\mathbb{R})$, $\frac{\partial F}{\partial x}$, $\frac{\partial G}{\partial x}$ are both bounded. Then the processes $\{U(t,\tau )\}$ on $C_{\gamma ,W^{1,r}(\Omega )}$ generated by the solution of Eq. (1) with $\phi \in C_{\gamma ,W^{1,r}(\Omega )}$ has the pullback attractors $\mathcal{A}_{C_{\gamma ,W^{1,r}(\Omega )}}$.

Proof

We divide the proof into three steps.

Step 1. Taking gradient operator to act on (1), we can obtain

$$\begin{aligned} \frac{\partial \nabla u}{\partial t}-\Delta \nabla u+\lambda \nabla u =& \frac{\partial F}{\partial x}+\frac{\partial F}{\partial u}\nabla u\bigl(t-\rho (t),x\bigr) + \int^{0}_{-\infty } \frac{\partial G}{\partial x}\,dz \\ & {} + \int^{0}_{-\infty } \frac{\partial G}{\partial u}\nabla u(t+z,x)\,dz+ \nabla g(t,x). \end{aligned}$$
(100)

Multiplying (100) by $\vert \nabla u \vert ^{r-2}\nabla u$ and integrating by parts, we get

$$\begin{aligned}& \frac{1}{r}\frac{d}{dt} \bigl\Vert \nabla u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )}+\frac{4(r-1)}{r ^{2}} \int_{\Omega } \bigl\vert \nabla \bigl( \bigl\vert \nabla u(t) \bigr\vert ^{\frac{r}{2}}\bigr) \bigr\vert ^{2}\,dx+ \int_{\Omega } \lambda \bigl\vert \nabla u(t) \bigr\vert ^{r}\,dx \\& \quad = \int_{\Omega } \frac{\partial F}{\partial x} \bigl\vert \nabla u(t) \bigr\vert ^{r-2} \nabla u(t)\,dx+ \int_{\Omega } \frac{\partial F}{\partial u}\nabla u\bigl(t- \rho (t),x\bigr) \vert \nabla u \vert ^{r-2}\nabla u\,dx \\& \quad\quad {} + \int_{\Omega } \int^{0}_{-\infty }\frac{\partial G}{\partial x} \bigl\vert \nabla u(t) \bigr\vert ^{r-2}\nabla u(t)\,dz\,dx+ \int_{\Omega } \int^{0}_{-\infty }\frac{ \partial G}{\partial u}\nabla u(t+z,x) \vert \nabla u \vert ^{r-2}\nabla u\,dz\,dx \\& \quad\quad {} + \int_{\Omega }\nabla g(t,x) \bigl\vert \nabla u(t) \bigr\vert ^{r-2}\nabla u(t)\,dx. \end{aligned}$$
(101)

By the same arguments as Lemma 4.1, we also obtain the process $\{U(t,\tau )\}$ generating by (100) has pullback absorbing sets in $C_{\gamma ,W^{1,r}(\Omega )}$.

Step 2. According to Theorem 15 [10], Eq. (1) has a pullback attractor $\mathcal{A}_{C_{\gamma ,H^{1}(\Omega )}}$. Hence, by the same arguments as Theorem 5.6, we also obtain the process $\{U(t,\tau )\}$ generating by Eq. (100) on $C_{\gamma ,L^{2}( \Omega )}$ is ω-limit compact.

Step 3. Combining step 1, step 2, and Lemma 6.1, as the proof of Theorem 5.6, we find that the process $\{U(t,\tau )\}$ generated by Eq. (100) on $C_{\gamma ,W^{1,r}(\Omega )}$ has pullback absorbing sets and is $\mathcal{D}$ pullback ω-limit compact. Thus, we know from Theorem 5.6 the process $\{U(t,\tau )\}$ generating by Eq. (1) has the pullback attractors $\mathcal{A}_{C_{\gamma ,W ^{1,r}(\Omega )}}$. □

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Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Nos. 11701244, 11772007, 11561060, 11661070, and 51468028), the Natural Science Foundation of Beijing (No. 1172002), the International Science and Technology Cooperation Program of China (No. 2014DFR61080).

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All authors read and approved the final manuscript. YR finished the manuscript and JL made the content correction and English language checking.

Correspondence to Yanping Ran.

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Keywords

  • Pullback attractor
  • Reaction–diffusion equation
  • Infinite delays
  • Nonautonomous equation