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Existence,uniqueness, and nonexistence of solutions to nonlinear diffusion equations with \(p(x,t)\)-Laplacian operator

Boundary Value Problems20162016:149

https://doi.org/10.1186/s13661-016-0657-9

©  Gao et al. 2016

  • Received: 19 August 2015
  • Accepted: 1 August 2016
  • Published:

Abstract

The aim of this paper is to deal with the existence and nonexistence of weak solutions to the initial and boundary value problem for \(u_{t}=\operatorname{div}(|\nabla u|^{p(x,t)-2}\nabla u+b(x,t)\nabla u)+f(u)\). By constructing suitable function spaces and applying the method of Galerkin’s approximation as well as weak convergence techniques, the authors prove the existence of local solutions. Furthermore, we choose a suitable test-function, make integral estimates, and apply Gronwall’s inequality to prove the uniqueness of weak solutions. At the end of this paper, the authors construct a suitable energy functional, obtain a new energy inequality, and apply a convex method to prove the nonexistence of solutions. Especially, it is worth pointing out that the results are obtained with the assumption that \(p_{t}(x,t)\) is only negative and integrable, which is weaker than most of the other papers required.

Keywords

  • nonstandard growth condition
  • nonexistence of solutions
  • Galerkin’s approximation

1 Introduction

Consider the following initial and boundary value problem:
$$ \textstyle\begin{cases} u_{t}=\operatorname{div}(|\nabla u|^{p(x,t)-2}\nabla u+b(x,t)\nabla u)+f(u),&(x,t)\in\Omega\times(0,T):=Q_{T}, \\ u(x,t)=0,&(x,t)=\partial\Omega\times(0,T):=\Gamma_{T}, \\ u(x,0)=u_{0}(x),&x\in\Omega, \end{cases} $$
(1.1)
where \(\Omega\subset\mathbb{R}^{N}\) (\(N\geqslant1\)) is a bounded domain, ∂Ω is Lipschitz continuous, and f is a continuous function satisfying
$$ \bigl\vert f(s)\bigr\vert \leqslant a_{0}|s|^{\alpha-1}, \quad 0< a_{0}=\mathrm{constant}, 1< \alpha=\mathrm{constant}. $$
(1.2)
It will be assumed throughout the paper that the exponent \(p(x,t)\) is continuous in \(Q=\overline{Q_{T}}\) with logarithmic module of continuity:
$$\begin{aligned}& 1< p^{-}=\inf_{(x,t)\in Q}p(x,t)\leqslant p(x,t)\leqslant p^{+}=\sup_{(x,t)\in Q}p(x,t)< \infty, \end{aligned}$$
(1.3)
$$\begin{aligned}& \forall z=(x,t),\xi=(y,s)\in Q_{T},|z-\xi|< 1,\quad \bigl\vert p(z)-p(\xi)\bigr\vert \leqslant\omega\bigl(\vert z-\xi \vert \bigr), \end{aligned}$$
(1.4)
where
$$ \limsup_{\tau\rightarrow0^{+}}\omega(\tau)\ln\frac{1}{\tau}=C< +\infty, $$
and the coefficient \(b(x,t)\) is a Carathéodory function.
Model (1.1) proposed by Růžička may describe some properties of electro-rheological fluids which change their mechanical properties dramatically when an external electric field is applied [1, 2]. The variable exponent p in Model (1.1) is a function of the external electric field \(|E|^{2}\) which is subject to the quasi-static Maxwell equations
$$\operatorname{div}(\varepsilon_{0}\vec{E}+\vec{P})=0,\qquad \operatorname{Curl}(\vec{E})=0, $$
where \(\varepsilon_{0}\) is the dielectric constant in vacuum and the electric polarization P⃗ is linear in E⃗, i.e. \(\vec{P}=\lambda\vec{E}\). Another important application is the image processing where the anisotropy and nonlinearity of the diffusion operator are used to underline the borders of the distorted image and to eliminate the noise [3, 4]. For more physical background, the interested reader may refer to [4–17].

In the case when \(p(x,t)\) is a fixed constant, there have been many results about the existence, uniqueness, nonexistence, extinction of the solutions [10, 18, 19]. For nonconstant case, the authors of [7–9] and the authors of [17] studied the existence and uniqueness of weak solutions of the initial and Dirichlet boundary value problem with variable exponent of nonlinearity. Besides, the authors of [20] applied the differential and variational techniques to prove the existence of solutions when the exponent p only depends on the spatial variable. Motivated by the work above, we consider the existence and uniqueness of solutions to Problem (1.1). However, since the coefficient \(b(x,t)\) is degenerate or singular, it is natural to ask: which kind of conditions on the coefficient \(b(x,t)\) guarantees that Problem (1.1) admits a local solution? In our paper, we construct suitable function spaces and apply Galerkin’s method to prove the existence of weak solutions to Problem (1.1) with necessary uniform estimates and compactness argument. In addition, there exist some difficulties such as the failure of the monotonicity of the energy functional, the anisotropy of the diffusive operator and the gap between the norm and the modular, which make the methods used in [19] fail. In order to overcome such difficulties, we have to search a new technique or method. In this paper, by constructing a revised energy functional and combining a new energy estimate with convex method, we obtain the nonexistence of weak solutions when the exponents p is a function with respect to time and spatial variables. Especially, it is worth pointing out that the results are obtained with the assumption that \(p_{t}(x,t)\) is only negative and integrable which is weaker than those the most of the other papers required.

The outline of this paper is the following: In Section 2, we shall introduce the function spaces of Orlicz-Sobolev type, give the definition of the weak solution to the problem, and prove the existence of weak solutions with Galerkin’s method and the uniqueness of the solution. In Section 3, we establish sufficient condition of nonexistence of weak solutions of Problem (1.1) with the assumption that the exponent \(p(x,t)\) depends on the time and spatial variables.

2 Existence of local solutions

In this section, the existence of weak solutions will be studied. First of all, we introduce some Banach spaces:
$$\begin{aligned}& L^{p(x)}(\Omega)= \biggl\{ u(x)\Big|u \mbox{ is measurable in } \Omega, A_{p(\cdot)}(u)= \int_{\Omega}|u|^{p(\cdot)}\,dx< \infty \biggr\} , \\& \|u\|_{p(\cdot)}=\inf\bigl\{ \lambda>0,A_{p(\cdot)}(u/\lambda)\leqslant1 \bigr\} ; \\& W^{1,p(x)}(\Omega):=\bigl\{ u: u\in L^{p(\cdot)}(\Omega),|\nabla u|\in L^{p(\cdot)}(\Omega)\bigr\} , \\& \|u\|_{W^{1,p(\cdot)}(\Omega)}=\|u\|_{p(\cdot), \Omega} +\|\nabla u\|_{p(\cdot), \Omega}; \\& V_{t}(\Omega)= \bigl\{ u| u\in L^{2}(\Omega)\cap W^{1,1}_{0}(\Omega), |u|^{p(x,t)}\in L^{1}( \Omega),|\nabla u|^{p(x,t)}\in L^{1}(\Omega) \bigr\} , \\& \|u\|_{V_{t}(\Omega)}=\|u\|_{2, \Omega} +\|\nabla u\|_{p(\cdot), \Omega}; \\& H(Q_{T})= \bigl\{ u:[0,T]\mapsto V_{t}(\Omega)| u\in L^{2}(Q_{T}), |\nabla u|\in L^{p(x,t)}(Q_{T}), u=0 \mbox{ on } \Gamma_{T} \bigr\} , \\& \|u\|_{H(Q_{T})}=\|u\|_{2, Q_{T}} +\|\nabla u\|_{p(\cdot), Q_{T}}, \end{aligned}$$
and denote by \(H'(Q_{T})\) the dual of \(H(Q_{T})\) with respect to the inner product in \(L^{2}(Q_{T})\). From [11], we know that condition (1.4) implies that \(M\overset{\Delta}{=}\{u: u\in W^{1,p(x)}(\Omega),u=0\mbox{ on } \partial\Omega\}\) is equivalent to \(W_{0}^{1,p(x)}(\Omega)\) (the closure of \(C^{\infty}_{0}(\Omega)\) in \(W^{1,p(x)}(\Omega)\)).

Lemma 2.1

[11, 12]

For any \(u\in L^{p(x)}(\Omega)\),
$$\begin{aligned} (1)&\quad \|u\|_{p(\cdot)}< 1\ (=1;>1)\quad \Leftrightarrow\quad A_{p(\cdot)}(u)< 1 \ (=1;>1); \\ (2)&\quad \|u\|_{p(\cdot)}< 1\quad \Rightarrow\quad \|u\|^{p^{+}}_{p(\cdot)} \leqslant A_{p(\cdot)}(u)\leqslant\|u\|^{p^{-}}_{p(\cdot)}; \\ &\quad \|u\|_{p(\cdot)}\geqslant1 \quad \Rightarrow\quad \|u \|^{p^{-}}_{p(\cdot)}\leqslant A_{p(\cdot)}(u)\leqslant\|u \|^{p^{+}}_{p(\cdot)}; \\ (3)&\quad \|u\|_{p(\cdot)}\rightarrow0\quad \Leftrightarrow\quad A_{p(\cdot)}(u)\rightarrow0; \\ &\quad \|u\|_{p(\cdot)}\rightarrow\infty\quad \Leftrightarrow\quad A_{p(\cdot)}(u)\rightarrow\infty. \end{aligned}$$

Lemma 2.2

[11, 12] (Hölder’s inequality)

For any \(u\in L^{p(x)}(\Omega)\) and \(v\in L^{q(x)}(\Omega)\) with \(q(x)=\frac{p(x)}{p(x)-1}\),
$$ \biggl\vert \int_{\Omega}uv \,dx\biggr\vert \leqslant\biggl( \frac{1}{p^{-}}+\frac{1}{q^{-}}\biggr) \|u\|_{p(\cdot)}\|v\|_{q(\cdot)} \leqslant2\|u\|_{p(\cdot)}\|v\|_{q(\cdot)}. $$

Because of the degeneracy, Problem (1.1) does not admit classical solutions in general. So we introduce weak solutions in the following sense.

Definition 2.1

A function \(u(x,t)\in H(Q_{T})\cap L^{\infty}(0,T;L^{2}(\Omega))\), \(b(x,t)|\nabla u|\in L^{2}(0,T;L^{2}(\Omega))\) is called a weak solution of Problem (1.1) if for every test-function
$$\xi\in\mathcal{Z}\overset{\Delta}{\equiv}\bigl\{ \eta(z):\eta\in H(Q_{T})\cap L^{2}\bigl(0,T;H^{1}_{0}( \Omega)\bigr),\eta_{t}\in H'(Q_{T})\bigr\} , $$
and every \(t_{1},t_{2}\in[0,T]\) the following identity holds:
$$ \int^{t_{2}}_{t_{1}} \int_{\Omega} \bigl[u\xi_{t}- \bigl(b(x,t)\nabla u+| \nabla u|^{p(x,t)-2}\nabla u \bigr)\nabla \xi+f(u)\xi \bigr]\,dx\,dt= \int_{\Omega}u\xi\,dx|^{t_{2}}_{t_{1}}. $$
(2.1)

Remark 2.1

On one hand, \(u\in H(Q_{T})\), \(\xi\in\mathcal{Z}\), implies that
$$\begin{aligned}& \int^{t_{2}}_{t_{1}} \int_{\Omega}u\xi_{t}\,dx\,dt< \infty, \\& \int^{t_{2}}_{t_{1}} \int_{\Omega}|\nabla u|^{p(x,t)-2}\nabla u\nabla \xi\,dx\,dt< \infty, \\& \int_{\Omega}u\xi\,dx|^{t_{2}}_{t_{1}}< \infty. \end{aligned}$$
On the other hand,
$$\begin{aligned}& u\in L^{\alpha}(Q_{T}),\quad \bigl\vert f(u)\bigr\vert \leqslant a_{0}|u|^{\alpha-1} \quad \mbox{and} \quad \xi\in L^{\alpha}(Q_{T})\quad \Longrightarrow\quad \int^{t_{2}}_{t_{1}} \int_{\Omega}f(u)\xi\,dx\,dt< \infty; \\& b(x,t)|\nabla u|\in L^{2}\bigl(0,T;L^{2}(\Omega)\bigr), \qquad \xi\in H(Q_{T})\quad \Longrightarrow\quad \int^{t_{2}}_{t_{1}} \int_{\Omega}b(x,t)\nabla u\nabla\xi\,dx\,dt< \infty, \end{aligned}$$
which imply that the definition of weak solutions is well defined.

The main theorem in this section is the following.

Theorem 2.1

Suppose that the continuous function \(f(s)\) and the exponents \(p(x,t)\), α satisfy conditions (1.2)-(1.4). If the following conditions hold:
$$\begin{aligned} (\mathrm{H}_{1}) &\quad 2< p^{-}< p^{+}< \max\biggl\{ N,\frac{Np^{-}}{N-p^{-}}\biggr\} ,\qquad 2< \alpha< p^{-}; \\ (\mathrm{H}_{2})&\quad u_{0}\in L^{2}(\Omega), \qquad b(x,t)\in L^{\frac{p^{-}}{p^{-}-2}}\bigl(0,\infty;L^{\frac{p^{-}}{p^{-}-2}}(\Omega)\bigr), \end{aligned}$$
then Problem (1.1) has at least one weak solution \(u(x,t)\in H(Q_{T})\cap L^{\infty}(0,T;L^{2}(\Omega))\), \(b(x,t)|\nabla u|\in L^{2}(0,T;L^{2}(\Omega))\).

Due to \(p^{-}>2\) and \(b(x,t)\in L^{\frac{p^{-}}{p^{-}-2}}(0,\infty;L^{\frac{p^{-}}{p^{-}-2}}(\Omega))\), the term \(b(x,t)|\nabla u|^{2}\) can be controlled by the nonlinear diffusion term \(|\nabla u|^{p(x,t)}\), and then one may follow the lines of the proof of Theorem 3.1(a) in [6] or Theorem 4.1(a) of Chapter 4 in [7] to complete the rest of the proof.

Corollary 2.1

Let the conditions of Theorem  2.1 be fulfilled, then the solution \(u\in H(Q_{T})\) to Problem (1.1) satisfies the identity
$$\begin{aligned}& \iint_{Q_{T}}u_{t}\xi\,dx+ \iint_{Q_{T}}\bigl[|\nabla u|^{p(x,t)-2}\nabla u\nabla \xi+b(x,t)\nabla u\nabla\xi-f(u)\xi \bigr]\,dx\,dt \\& \quad =0, \quad \forall\xi\in \mathcal{Z}. \end{aligned}$$
(2.2)

When \(b(x,t)\geqslant0\), we follow the line of the proof of Theorem 5.1 in [6] to obtain the following theorem.

Theorem 2.2

Suppose that the conditions in Theorem  2.1 are fulfilled and \(b(x,t)\geqslant0\), then the bounded solution of Problem (1.1) is unique provided that the following condition holds
$$ (\mathrm{H}_{3})\quad \textit{the function }f(s)\textit{ is decreasing in }s\in\mathbb{R}. $$

Furthermore, we have the following comparison theorem.

Corollary 2.2

(Comparison principle)

Let \(u,v\in H(Q_{T})\cap L^{2}(0,T;H^{1}_{0}(\Omega))\) be two bounded weak solutions of Problem (1.1) such that \(u(x,0)\leqslant v(x,0)\) a.e. in Ω. If the nonlinearity exponents and the function \(f(s)\) satisfy the conditions of Theorem  3.1, then \(u(x,t)\leqslant v(x,t)\) a.e. in \(Q_{T}\).

3 Nonexistence of global weak solutions

In this section, we concentrate on the study of nonexistence of weak solutions to Problem (1.1). For convenience, we first state that the function \(f(s)\) and the coefficient \(b(x,t)\) satisfy the following conditions:
$$\begin{aligned}& b(x,t)\geqslant0,\qquad b_{t}(x,t)\leqslant0,\quad \forall(x,t)\in Q_{T}; \end{aligned}$$
(3.1)
$$\begin{aligned}& f(u)\in C(R),\qquad f(u)u- p^{+}G(u)\geqslant0,\quad \forall u\in \mathbb{R}, \end{aligned}$$
(3.2)
with \(G(u)=\int^{u}_{0}f(s)\,ds\). Before stating the main results, we give the definition of global solutions.

Definition 3.1

A function \(u(x,t)\) is called a global solution to Problem (1.1) if \(\forall T>0\) the following property holds:
$$\sup_{t\in(0,T)}\bigl\Vert u(x,t)\bigr\Vert _{L^{\infty}(\Omega)}< + \infty. $$
Otherwise, we say that Problem (1.1) does not admit global weak solutions.

First, we consider the case \(p(x,t)\equiv p(x)\). Our main result is as follows.

Theorem 3.1

Assume that \(u(x,t)\in H(Q_{T})\cap L^{\infty}(0,T;L^{2}(\Omega))\), \(b(x,t)|\nabla u|\in L^{2}(0,T;L^{2}(\Omega))\) is the local solution to Problem (1.1). If (3.1) and (3.2) are fulfilled and \(u_{0}\in W^{1,p^{(}x)}_{0}(\Omega)\), \(p^{+}>2\), such that
$$\begin{aligned} \int_{\Omega}G(u_{0})\,dx >& \int_{\Omega}\frac{b(x,0)}{2}|\nabla u_{0}|^{2} \,dx+ \int_{\Omega}\frac{1}{p(x)}|\nabla u_{0}|^{p(x)} \,dx \\ &{}+\frac{4(p^{+}-1)}{Tp^{+}(p^{+}-2)^{2}} \int_{\Omega }|u_{0}|^{2}\,dx. \end{aligned}$$
(3.3)
Then there exists a \(T^{*}\in(0,T]\) such that
$$\lim_{t\rightarrow T^{*-}}\bigl\Vert u(\cdot,t)\bigr\Vert _{\infty,\Omega}=+\infty. $$

To prove Theorem 3.1, we need the following lemma.

Lemma 3.1

Assume that \(u\in H(Q_{T})\) is the solution to Problem (1.1), then \(u(x,t)\) satisfies the following relation:
$$\begin{aligned}& \int_{\Omega}\frac{1}{p(x)}\bigl\vert \nabla u(x,t)\bigr\vert ^{p(x)}\,dx+ \int_{\Omega}\frac{b(x,t)}{2}\bigl\vert \nabla u(x,t)\bigr\vert ^{2}\,dx- \int_{\Omega}G\bigl(u(x,t)\bigr)\,dx \\& \qquad {}+ \int^{t}_{0} \int_{\Omega }(u_{\tau})^{2}\,dx\,d\tau - \int^{t}_{0} \int_{\Omega}\frac{b_{\tau} \vert \nabla u\vert ^{2}}{2}\,dx\,d\tau \\& \quad = \int_{\Omega}\frac{1}{p(x)}\bigl\vert \nabla u_{0}(x)\bigr\vert ^{p(x)}\,dx+ \int_{\Omega}\frac{b(x,0)}{2}\bigl\vert \nabla u_{0}(x)\bigr\vert ^{2}\,dx- \int_{\Omega}G\bigl(u_{0}(x)\bigr)\,dx. \end{aligned}$$
(3.4)

Proof

Following the lines of the proof of Lemma 3.1 and Theorem 6.1 in [7], we know that \(u_{t}\in L^{2}(Q_{T})\). Noting that
$$\begin{aligned}& \frac{\partial}{\partial t}\biggl(\frac{|\nabla u|^{p(x)}}{p(x)}\biggr)=|\nabla u|^{p(x)-2} \nabla u\nabla u_{t},\qquad \frac{\partial}{\partial t}\bigl(G\bigl(u(x,t)\bigr) \bigr)=f\bigl(u(x,t)\bigr)u_{t}, \\& \frac{\partial}{\partial t}\biggl(b(x,t)\frac{|\nabla u|^{2}}{2}\biggr)=b(x,t)\nabla u\nabla u_{t}+b_{t}\frac{|\nabla u|^{2}}{2} \end{aligned}$$
and using the idea of the proof of Lemma 1 in [8], we arrive at the relation
$$ \frac{d}{dt} \int_{\Omega} \biggl(\frac{|\nabla u|^{p(x)}}{p(x)}+\frac{b(x,t)}{2}\bigl|\nabla u(x)\bigr|^{2}-G\bigl(u(x,t)\bigr) \biggr)\,dx- \int_{\Omega}\frac{b_{t}|\nabla u|^{2}}{2}\,dx=- \int_{\Omega}u_{t}^{2}\,dx. $$

After integrating over \((0,t)\), it is obvious that Lemma 3.1 holds. □

Proof of Theorem 3.1

Let
$$\begin{aligned}& \beta=\frac{2}{T(p^{+}-2)^{2}} \int_{\Omega}u_{0}^{2}\,dx,\qquad t_{0}=\frac {T(p^{+}-2)}{2}, \\& K(t)=\frac{1}{2} \int^{t}_{0} \int_{\Omega}u^{2}\,dx\,d\tau+(T-t) \int _{\Omega}\frac{1}{2}u_{0}^{2}\,dx+ \beta(t+t_{0})^{2}. \end{aligned}$$
Clearly
$$\begin{aligned}& K'(t) =\frac{1}{2} \int_{\Omega}u^{2}\,dx\,dt-\frac{1}{2} \int_{\Omega }u_{0}^{2}\,dx+2 \beta(t+t_{0}) \\& \hphantom{K'(t)}= \iint_{Q_{t}} \bigl(-b(x,\tau)|\nabla u|^{2}-|\nabla u|^{p(x)}+f(u)u \bigr)\,dx\,dt+2\beta(t+t_{0}); \\& K''(t) = \int_{\Omega} \bigl(-b(x,\tau)|\nabla u|^{2}-|\nabla u|^{p(x)}+f(u)u \bigr)\,dx+2\beta. \end{aligned}$$
By Hölder’s inequality, we have
$$\begin{aligned} \frac{1}{2} \int_{\Omega}u^{2}\,dx\,dt-\frac{1}{2} \int_{\Omega }u_{0}^{2}\,dx &= \frac{1}{2}\biggl\vert \int^{t}_{0} \int_{\Omega}\bigl(u^{2}\bigr)_{\tau}\,dx\,d\tau \biggr\vert \\ &\leqslant\biggl( \int^{t}_{0} \int_{\Omega}u^{2}\,dx\,d\tau\biggr)^{1/2} \biggl( \int ^{t}_{0} \int_{\Omega}|u_{\tau}|^{2}\,dx\,d\tau \biggr)^{1/2}. \end{aligned}$$
Thus by Schwarz’s inequality and the definition of \(K(t)\), we have
$$\begin{aligned} \bigl(K'(t)\bigr)^{2} \leqslant& \biggl( \int^{t}_{0} \int_{\Omega}u^{2}\,dx\,d\tau \biggr) \biggl( \int ^{t}_{0} \int_{\Omega}u_{\tau}^{2}\,dx\,d\tau \biggr) +4 \beta^{2}(t+t_{0})^{2} \\ &{} +4\beta(t+t_{0}) \biggl( \int^{t}_{0} \int_{\Omega}u^{2}\,dx\,d\tau \biggr)^{1/2} \biggl( \int^{t}_{0} \int_{\Omega}u_{\tau}^{2}\,dx\,d\tau \biggr)^{1/2} \\ \leqslant& \biggl( \int^{t}_{0} \int_{\Omega}u^{2}\,dx\,d\tau \biggr) \biggl( \int ^{t}_{0} \int_{\Omega}u_{\tau}^{2}\,dx\,d\tau \biggr) +4 \beta^{2}(t+t_{0})^{2} \\ &{} +2 \int^{t}_{0} \int_{\Omega}u_{\tau}^{2}\,dx\,d\tau \biggl[(T-t) \int _{\Omega}\frac{1}{2}u_{0}^{2}\,dx+ \beta(t+t_{0})^{2} \biggr] \\ &{} +4\beta^{2}(t+t_{0})^{2} \biggl( \int^{t}_{0} \int_{\Omega}\frac {1}{2}u^{2}\,dx\,d\tau \biggr) \biggl((T-t) \int_{\Omega}\frac{1}{2}u_{0}^{2}\,dx+ \beta(t+t_{0})^{2} \biggr)^{-1} \\ \leqslant& K(t) \biggl(2 \int^{t}_{0} \int_{\Omega}u_{\tau}^{2}\,dx\,d\tau +4\beta \biggr). \end{aligned}$$
Therefore by Lemma 3.1, we obtain the following inequality:
$$\begin{aligned}& K(t)K''(t)-\frac{p^{+}}{2} \bigl(K'(t)\bigr)^{2} \\& \quad \geqslant K(t) \biggl( \int_{\Omega }b(x,t) \biggl(\frac{p^{+}}{2}-1\biggr)|\nabla u|^{2}\,dx+ \int_{\Omega}\biggl(\frac{p^{+}}{p(x)}-1\biggr)|\nabla u|^{p(x)}\,dx \\& \qquad {}+ \int_{\Omega} \bigl(uf(u)-p^{+}G(u) \bigr)\,dx \biggr)-2 \beta\bigl(p^{+}-1\bigr) \\& \qquad {}+\frac{p^{+}}{2} \int_{\Omega} \biggl[2G(u_{0})-b(x,0)|\nabla u_{0}|^{2}-\frac{2}{p(x)}|\nabla u_{0}|^{p(x)} \biggr]\,dx \\& \qquad {}-\frac{p^{+}}{2} \int^{t}_{0} \int_{\Omega}\frac{b_{t}|\nabla u|^{2}}{2}\,dx\,dt. \end{aligned}$$
(3.5)
Noticing \(p^{+}>2\), \(K(t)>0\), we conclude from (3.1), (3.2), (3.3) that
$$ K(t)K''(t)-\frac{p^{+}}{2}\bigl(K'(t) \bigr)^{2}\geqslant0,\quad \mbox{for }t\in(0,T), $$
which implies
$$ \bigl(K^{1-\frac{p^{+}}{2}}(t)\bigr)''\leqslant0,\quad \mbox{for }t\in(0,T). $$
Noting that \(K^{1-\frac{p^{+}}{2}}(0)>0\), \((K^{1-\frac{p^{+}}{2}})'(0)\leqslant0\), then
$$ K^{1-\frac{p^{+}}{2}}\bigl(T^{*}\bigr)=0, \quad \mbox{for some }T^{*}\in\biggl(0,\frac{-K^{1-\frac {p^{+}}{2}}(0)}{(K^{1-\frac{p^{+}}{2}})'(0)}\biggr). $$
Here
$$ \frac{-K^{1-\frac{p^{+}}{2}}(0)}{(K^{1-\frac{p^{+}}{2}})'(0)}= \frac{T\int_{\Omega}u^{2}_{0}\,dx+2\beta t_{0}^{2}}{2(p^{+}-2)\beta t_{0}^{2}}\leqslant T. $$
Thus, \(0< T^{*}\leqslant T\) and
$$\lim_{t\rightarrow T^{*-}}\bigl\vert u(\cdot,t)\bigr\vert _{\infty,\Omega}=+\infty. $$

This completes the proof of Theorem 3.1. □

Next, we consider the case when p is dependent of t. Before stating the conclusion, we first give a useful lemma.

Lemma 3.2

Assume \(u_{0}\in W^{1,p(x,0)}_{0}(\Omega)\), \(p^{+}>2\), \(p_{t}\leqslant0\). Then the solution of Problem (1.1) satisfies
$$\begin{aligned}& \int_{\Omega}\frac{1}{p(x,t)}\bigl\vert \nabla u(x,t)\bigr\vert ^{p(x,t)}\,dx+ \int_{\Omega}\frac{b(x,t)}{2}\bigl\vert \nabla u(x,t)\bigr\vert ^{2}\,dx- \int_{\Omega}G\bigl(u(x,t)\bigr)\,dx \\& \qquad {}+ \int^{t}_{0} \int_{\Omega }\vert u_{\tau} \vert ^{2}\,dx\,d \tau - \int^{t}_{0} \int_{\Omega}\frac{b_{\tau} \vert \nabla u\vert ^{2}}{2}\,dx\,d\tau \\& \quad \leqslant \int_{\Omega}\frac{1}{p(x,0)}\bigl\vert \nabla u_{0}(x)\bigr\vert ^{p(x,0)}\,dx+ \int_{\Omega}\frac{b(x,0)}{2}\bigl\vert \nabla u_{0}(x)\bigr\vert ^{2}\,dx- \int_{\Omega}G\bigl(u_{0}(x)\bigr)\,dx \\& \qquad {}+ \int_{\Omega} \biggl(\frac{1}{p(x,t)}-\frac{1}{p(x,0)} \biggr) \,dx. \end{aligned}$$
(3.6)

Proof

Following the lines of the proof of Lemmas 3.1 and Lemma 6.1 of [7], we know \(u_{t}\in L^{2}(Q_{T})\) and
$$\begin{aligned}& \frac{\partial}{\partial t}\biggl(\frac{|\nabla u|^{p(x,t)}}{p(x,t)}\biggr)=|\nabla u|^{p(x,t)-2} \nabla u\nabla u_{t}+\frac{p_{t}}{p^{2}}|\nabla u|^{p(x,t)} \bigl( \ln|\nabla u|^{p(x,t)}-1 \bigr), \\& \frac{\partial}{\partial t}\bigl(G\bigl(u(x,t)\bigr)\bigr)=f\bigl(u(x,t) \bigr)u_{t},\qquad \frac{\partial}{\partial t}\biggl(b(x,t)\frac{|\nabla u|^{2}}{2} \biggr)=b(x,t)\nabla u\nabla u_{t}+b_{t}\frac{|\nabla u|^{2}}{2}. \end{aligned}$$
On one hand, a simple analysis shows that
$$\begin{aligned}& \frac{d}{dt} \int_{\Omega}\biggl(\frac{\vert \nabla u\vert ^{p(x,t)}}{p(x,t)} +\frac{b(x,t)}{2}\bigl\vert \nabla u(x)\bigr\vert ^{2}-G\bigl(u(x,t)\bigr)\biggr)\,dx- \int_{\Omega}\frac{b_{t}\vert \nabla u\vert ^{2}}{2}\,dx \\& \quad = \int_{\Omega} \biggl[-u_{t}^{2}+ \frac {p_{t}}{p^{2}}\vert \nabla u\vert ^{p(x,t)} \bigl(\ln \vert \nabla u\vert ^{p(x,t)}-1 \bigr) \biggr]\,dx. \end{aligned}$$
(3.7)
On the other hand, we apply the condition \(p_{t}\leqslant0\) to obtain
$$\begin{aligned}& \int_{\{|\nabla u|^{p}\leqslant e\}} \frac{|\nabla u|^{p(x,t)}}{p^{2}(x,t)} \bigl(\ln|\nabla u|^{p(x,t)}-1 \bigr)p_{t}(x,t)\,dx \\& \quad \leqslant \int_{\{|\nabla u|^{p}\leqslant e\}} \frac{-p_{t}(x,t)}{p^{2}(x,t)}\,dx\leqslant \int_{\Omega}\frac{-p_{t}(x,t)}{p^{2}(x,t)}\,dx. \end{aligned}$$
(3.8)
The second inequality above follows from
$$ -\frac{1}{e}\leqslant s\ln s\leqslant0,\qquad 0\leqslant s\leqslant1. $$
Lemma 3.2 follows from (3.7) and (3.8). □

Our main result is as follows.

Theorem 3.2

Suppose that (3.1) and (3.2) hold and \(p^{+}>2\), \(p_{t}\leqslant0\). If \(u_{0}\in W^{1,p(x,0)}_{0}(\Omega)\) satisfies
$$\begin{aligned} \int_{\Omega}G(u_{0})\,dx >& \int_{\Omega}\frac{b(x,0)}{2}|\nabla u_{0}|^{2} \,dx+ \int_{\Omega} \frac{1}{p(x,0)}|\nabla u_{0}|^{p(x,0)} \, dx \\ &{} +\frac{4(p^{+}-1)}{Tp^{+}(p^{+}-2)^{2}} \int_{\Omega}|u_{0}|^{2}\, dx+ \int_{\Omega} \biggl(\frac{2}{p^{-}}-\frac{1}{p(x,0)} \biggr) \,dx, \end{aligned}$$
(3.9)
then there exists \(T^{*}\in(0,T]\) such that
$$\lim_{t\rightarrow T^{*-}}\bigl\vert u(\cdot,t)\bigr\vert _{\infty,\Omega}=+\infty. $$

Proof

We argue by contradiction. Define
$$\begin{aligned}& \beta=\frac{2}{T(p^{+}-2)^{2}} \int_{\Omega}u_{0}^{2}\,dx,\qquad t_{0}=\frac {T(p^{+}-2)}{2}, \\& K(t)=\frac{1}{2} \int^{t}_{0} \int_{\Omega}u^{2}\,dx\,d\tau+(T-t) \int _{\Omega}\frac{1}{2}u_{0}^{2}\,dx+ \beta(t+t_{0})^{2}. \end{aligned}$$
It is easy to verify that
$$ \begin{aligned} &K'(t)=\frac{1}{2} \int_{\Omega}u^{2}\,dx\,dt-\frac{1}{2} \int_{\Omega }u_{0}^{2}\,dx+2 \beta(t+t_{0}) \\ &\hphantom{K'(t)}= \iint_{Q_{t}} \bigl(-b(x,\tau)|\nabla u|^{2}-|\nabla u|^{p(x,t)}+f(u)u \bigr)\,dx\,dt+2\beta(t+t_{0}); \\ &K''(t)= \int_{\Omega} \bigl(-b(x,\tau)|\nabla u|^{2}-|\nabla u|^{p(x,t)}+f(u)u \bigr)\,dx+2\beta; \\ &\bigl(K'(t)\bigr)^{2} \leqslant K(t) \biggl(2 \int^{t}_{0} \int_{\Omega}u_{\tau}^{2}\,dx\,d\tau +4\beta \biggr). \end{aligned} $$
(3.10)
Therefore by Lemma 3.2 and (3.10), we obtain the following inequality:
$$\begin{aligned}& K(t)K''(t)-\frac{p^{+}}{2} \bigl(K'(t)\bigr)^{2} \\& \quad \geqslant K(t) \biggl[ \int_{\Omega }b(x,t) \biggl(\frac{p^{+}}{2}-1\biggr)|\nabla u|^{2}\,dx+ \int_{\Omega}\biggl(\frac{p^{+}}{p(x,t)}-1\biggr)|\nabla u|^{p(x,t)}\, dx \\& \qquad {} + \int_{\Omega} \bigl(uf(u)-p^{+}G(u) \bigr)\,dx-2\beta \bigl(p^{+}-1\bigr) \\& \qquad {} +\frac{p^{+}}{2} \int_{\Omega} \biggl[2G(u_{0})-b(x,0)|\nabla u_{0}|^{2}-\frac{2}{p(x,0)}|\nabla u_{0}|^{p(x,0)} \biggr]\,dx \\& \qquad {} -\frac{p^{+}}{2} \int^{t}_{0} \int_{\Omega}\frac{b_{\tau}|\nabla u|^{2}}{2}\,dx\,d\tau+\frac{p^{+}}{2} \int_{\Omega} \biggl(\frac {1}{p(x,t)}-\frac{1}{p(x,0)} \biggr) \,dx \biggr]. \end{aligned}$$
(3.11)
In the rest of the proof, we follow the lines of the proof of Theorem 3.1 to finish the proof of this theorem. □

At the end of this paper, we give an example to illustrate that the condition on \(p_{t}(x,t)\) is weaker than that of [8, 9].

Example 3.1

Set \(p(x,y,z,t)=\frac{\sqrt{t}\cos x}{100}+\frac{5}{2}\), \(x\in(\frac{\pi}{2},\pi)\), \(y,z\in(0,\frac{\pi}{2})\), \(0< t<1\). A simple computation shows that
$$\begin{aligned}& p_{t}(x,y,z,t)=\frac{\cos x}{200\sqrt{t}}< 0, \\& \int_{0}^{1} \int_{\frac{\pi }{2}}^{\pi} \iint_{0}^{\frac{\pi}{2}}\bigl\vert p_{t}(x,y,z,t) \bigr\vert \, dx\, dy\, dz\, dt=\frac{\pi ^{2}}{400},\quad p_{t}(x,y,z,t) \notin L^{\infty}. \end{aligned}$$

Declarations

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11271154) and by the 985 program of Jilin University.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Science, Changchun University of Science and Technology, Changchun, 130022, P.R. China
(2)
School of Mathematics, Jilin University, Changchun, 130012, P.R. China

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