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Global existence and stability of a class of nonlinear evolution equations with hereditary memory and variable density
Boundary Value Problems volume 2019, Article number: 37 (2019)
Abstract
In this paper, we consider the initial boundary value problem of nonlinear evolution equation with hereditary memory, variable density, and external force term
Under suitable assumptions, we prove the existence of a global solution by means of the Galerkin method, establish the exponential stability result by using only one simple auxiliary functional, and give the polynomial stability result.
1 Introduction
In this paper, we are concerned with the following problem:
where Ω is a bounded domain of \(\mathbb{R}^{n}\) (\(n\geq 1\)) with smooth boundary ∂Ω, ρ is a positive constant, and \(\gamma \geq 0\). We prove the existence of a global solution by means of the Galerkin method and establish the exponential stability under suitable assumptions by using a simpler auxiliary functional than that in [1]. We also show the polynomial stability under suitable conditions.
Partial differential equations in viscoelastic materials have important physical background and important mathematical significance. The viscous effects are described and characterized by an integral term, and the integral term indicates a dissipative effect. For mathematical analysis on the motions of evolution equations with memory, we refer to [8, 32]. Problem (1.1) is related to the equations
which have several modeling features. If \(f(u_{t})\) is a constant, Eq. (1.2) has been used to model extensional vibrations of thin rods (see [27, Ch. 20]) and it differs from D’Alembert’s wave equation because of \(\Delta u_{tt}\), which is not a damping term. On the contrary, \(\Delta u_{tt}\) increases the energy functional. If \(f(u_{t})\) is not a constant, Eq. (1.2) shows that the density of materials depends on the velocity \(u_{t}\).
In the past ten years, several authors studied the homogeneous Dirichlet boundary value problem for the following model with memory (starting from the zero moment) and variable density:
in a bounded domains \(\varOmega \subset \mathbb{R}^{n}\). Cavalcanti et al. [2] considered the model with integral dissipation and strong damping
Assuming that \(0<\rho \leq \frac{2}{n-2}\) if \(n \geq 3\) or \(\rho >0\) if \(n = 1,2\) and that \(g(t)\) decays exponentially, they obtained the global existence of a solution for \(\gamma \geq 0\) and the uniform exponential decay of the energy for \(\gamma > 0\). Cavalcanti et al. [3] considered this model and proved intrinsic decays for large classes of relaxation kernels described by the inequality \(g'+ H(g)\leq 0\) with convex function H. Han and Wang [11] considered the equation with integral dissipation and linear damping
They proved the global existence and exponential decay when g is decaying exponentially by introducing two auxiliary functionals. Han and Wang [12] established the general decay of energy for the equation with integral dissipation and nonlinear damping
by introducing two auxiliary functionals. Messaoudi and Tatar [29, 30] considered the equation only with integral dissipation
Under some assumptions on g, they obtained exponential and polynomial decay rates. Messaoudi and Tatar [28] studied the equation with external force term and only with integral dissipation
By introducing a new functional and using potential well method they showed that there exists an appropriate set S (called a stable set) such that if the initial datum is in S, then the solution continues to live there forever. They also showed that the solution goes to zero with an exponential or polynomial rate depending on the decay rate of the relaxation function g. Liu [26] considered (1.3) and proved that, for certain class of relaxation functions and certain initial data in the stable set, the decay rate of the solution energy is similar to that of the relaxation function. Conversely, for certain initial data in the unstable set, there are solutions that blow up in finite time.
Now, we list some important literature on the nonlinear evolution equation with hereditary memory and variable density. Araújo et al. [1] considered the equation with integral dissipation in infinite interval
where Ω is a bounded domain of \(\mathbb{R}^{n}\) (\(n\geq 1\)) with smooth boundary ∂Ω. They established the uniqueness of the solution, exponential decay, and global attractors. However, the existence of a solution is not given in detail, two auxiliary functionals are introduced to prove the exponential decay result, and the polynomial decay result is not given. Conti et al. [5] established an existence, uniqueness, and continuous dependence result for weak solutions to the nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain
with Dirichlet boundary conditions. In particular, the parameter ρ belongs to the interval \([0,4]\), the value 4 is critical for the Sobolev embeddings, whereas f can reach the critical polynomial order 5. Lately, Conti et al. [4] studied the nonlinear viscoelastic equation
and showed that the sole weak dissipation given by the memory term is enough to ensure the existence and optimal regularity of the global attractor \(\mathcal{A}_{\rho }\) for \(\rho <4\) and critical nonlinearity f.
In recent years, Fatori et al. [9] studied long-time behavior of a class of thermoelastic plates with nonlinear strain and long memory; the main result establishes the existence of global and exponential attractors for the strongly damped problem through a stabilizability inequality. In addition, for the weakly damped problem, they establish the exponential stability of its Galerkin semiflows. Li et al. [13,14,15] proved the existence uniqueness, uniform energy decay rates, and limit behavior of the solution to the nonlinear viscoelastic Marguerre–von Kármán shallow shells system. The global existence uniqueness and decay estimates for nonlinear viscoelastic equation with boundary dissipation were given in [16, 17, 19, 22,23,24,25]. The authors in [10, 18, 20, 21] studied the blowup phenomenon for some evolution equations. Du and Li [6, 7] proved the integrability and regularity of the solution to some equations.
In this paper, we study the equation with hereditary memory (\(u_{0}(x,t)\), \(t\leq 0\)) and variable density
that is,
which can be rewritten as
This equation inspires us to define
which implies
and
Hence Eq. (1.4) can be rewritten as
Without loss of generality, we assume that \(\alpha -\int_{0}^{\infty }\mu (\tau)\,d\tau =1\). Then
where
The main contribution of this paper are: (a) the equation with hereditary memory, variable density, and external force term is representative; (b) the detailed construction process of the energy functional is given by an integration method; (c) we give a detailed proof of the existence for the solution; (d) the proof of the exponential decay result is simplified by introducing only one auxiliary functional; (e) the polynomial decay result is established.
The outline of this paper is as follows. In Sect. 2, we present the preliminaries and important our results. In Sects. 3–5, we prove the main Theorems 2.1–2.3, respectively.
2 Assumptions and the main results
In this paper, we assume that the following conditions \((A_{1})\)–\((A _{3})\) hold:
- \((A_{1})\) :
-
$$\begin{aligned} 0< \rho \leq \frac{2}{n-2} \quad \textit{if } n\geq 3;\qquad \rho >0 \quad \textit{if } n=1,2, \end{aligned}$$
which implies that
$$\begin{aligned} H_{0}^{1}(\varOmega)\hookrightarrow L^{2(\rho +1)}( \varOmega). \end{aligned}$$ - \((A_{2})\) :
-
\(f:\mathbb{R}\rightarrow \mathbb{R}\) and satisfies
$$\begin{aligned} \bigl\vert f(u)-f(v) \bigr\vert \leq c_{0} \bigl(1+ \vert u \vert ^{p}+ \vert v \vert ^{p} \bigr) \vert u-v \vert ,\quad u,v\in \mathbb{R}, \end{aligned}$$where \(c_{0}>0\) and
$$\begin{aligned} 0< p\leq \frac{2}{n-2} \quad \textit{if } n\geq 3;\qquad \rho >0 \quad \textit{if } n=1,2, \end{aligned}$$and
$$\begin{aligned} f(s)s\leq F(s)\leq 0,\quad \forall s\in \mathbb{R}, \end{aligned}$$where \(F(z)=\int_{0}^{z}f(\sigma)\,d\sigma \).
- \((A_{3})\) :
-
μ satisfies
$$\begin{aligned} \mu \in C^{1} \bigl(\mathbb{R}^{+} \bigr)\cap L^{1} \bigl(\mathbb{R}^{+} \bigr),\qquad 0\leq \mu (\tau)< \infty,\qquad \mu (0)>0,\qquad \mu (+\infty)=0 \end{aligned}$$with
$$\begin{aligned} \int_{0}^{\infty }\mu (\tau)\,d\tau =:k_{0}>0, \end{aligned}$$and there exists a constant \(k_{1}>0\) satisfying
$$\begin{aligned} \mu '(t)\leq -k_{1}\mu^{q}(t),\quad \forall t \in \mathbb{R}^{+}, 1\leq q< \frac{3}{2}. \end{aligned}$$
To consider the relative displacement η as a new function, we introduce the weighted \(L^{2}\)-space
which is a Hilbert space endowed with inner product
and norm
We introduce the notation
Remark 2.1
-
(1)
\(H_{0}^{1}(\varOmega)\hookrightarrow L^{r}( \varOmega)\) with
$$\begin{aligned} r:\textstyle\begin{cases} 2\leq r\leq \frac{2n}{n-2}, & n\geq 3, \\ \geq 2, & n=1,2, \end{cases}\displaystyle \end{aligned}$$which implies
$$\begin{aligned} \Vert \varphi \Vert _{r}\leq B \Vert \varphi \Vert _{2},\quad \forall \varphi \in H_{0}^{1}( \varOmega). \end{aligned}$$ -
(2)
From \((A_{2})\) we can easily get \(f(0)=0\).
-
(3)
The condition \(q<\frac{3}{2}\) is imposed to ensure that \(\int_{0}^{\infty }\mu^{2-q}(\tau)\,d\tau <\infty \). In fact, assumption (\(A_{3}\)) implies
$$\begin{aligned} \mu (t)\leq \frac{C_{1}}{(1+t)^{\frac{1}{q-1}}}, \qquad \frac{2-q}{q-1}>1, \end{aligned}$$and therefore \(\int_{0}^{\infty }\mu^{2-q}(\tau)\,d\tau <\infty \).
Give the initial data \((u_{0},u_{1},\eta_{0})\in \mathcal{H}\), a function \(\boldsymbol{z}=(u,u_{t},\eta)\in C([0,T],\mathcal{H})\) is a weak solution of problem (1.5) if it satisfies the initial condition \(\boldsymbol{z}(0)=(u_{0},u_{1},\eta_{0})\) and
for all \(w\in H_{0}^{1}(\varOmega),v\in \mathcal{M}\), and a.e. \(t\in [0,T]\).
Multiplying both sides of Eq. (1.5) by \(u_{t}\), integrating the resulting equation over Ω, and using the Green formula, we have
that is,
A direct computation and application of (1.5) show that
This computation inspires us to define an energy functional as follows:
and
Using \((A_{3})\) and (1.5), we have
Then
Theorem 2.1
Assume that conditions \((A_{1})\)–\((A_{3})\) hold and \(\gamma \geq 0\). If the initial data \((u_{0},u_{1},\eta_{0}) \in \mathcal{H}\), then for any \(T>0\), problem (1.5) has a weak solution
satisfying
Theorem 2.2
Assume that conditions \((A_{1})\)–\((A_{3})\) hold and \(\gamma >0\). If \(q=1\), then
where K and ν are positive constants.
Theorem 2.3
Assume that conditions \((A_{1})\)–\((A_{3})\) hold and \(\gamma >0\). If \(1< q<\frac{3}{2}\), then
where K is a positive constant.
3 Proof of Theorem 2.1
We study the equation
Let \(\{\omega_{j}\}^{\infty }_{j=1}\) be an orthogonal basis of \(H_{0}^{1}\) with \(\omega_{j}\) satisfying
By normalization we have \(\|\omega_{j}\|_{2}=1\) and write \(V_{k}= \operatorname{span}\{\omega_{1},\ldots,\omega_{k}\}\). For any given integer k, we consider the approximate solution
that satisfies
and, as \(k\rightarrow \infty \),
Here we denote by \((\cdot,\cdot)\) the inner product in \(L^{2}( \varOmega)\). Then (3.1) can be reduced to the second-order ODE system
According to the standard existence theory for ordinary differential equations, we infer that system (3.3) admits a solution \(c_{k}^{j}(t)\) in \([0,t_{m})\), where \(t_{m}>0\). Then we can obtain an approximate solution \(u_{k}(t)\) of (3.1) in \(V_{k}\) over \([0,t_{m})\), and the solution can be extended to \([0,T]\) for any given \(T>0\).
Multiplying (3.3) by \({c_{k}^{j}}'(t)\) and summing with respect to j, we conclude that
Simple calculations yield
Combining (3.4) and (3.6), we find
Integrating (3.7) over \((0,t)\) and noting (3.2), we obtain
where \(K_{1}\) is a constant independent of k. It follows from (3.8) and the Poincaré inequality that
Multiplying (3.1) by \({c_{k}^{j}}''(t) \) and summing with respect to j, we obtain
that is,
The right-hand side of (3.10) can be estimated as follows:
with \(k_{0}=\int_{0}^{\infty }\mu (\tau)\,d\tau \). Using (\(A_{2}\)), the Sobolev embedding theorem, and the Poincaré inequality, we have
that is,
From (3.14) and (3.18) we know that
Integrating (3.15) over \((0,t)\) (\(0< t\leq T\)) and noting (3.2) yield
Taking ε suitably small in (3.16), we can obtain the second estimate
which implies that
According to estimates (3.9) and (3.17), we infer that there exists a subsequence in \(\{u_{m}\}\) (denoted by the same symbol) such that
which, combined with the Aubin–Lions compactness lemma, implies
Using \((A_{2})\) and (3.19), we get
From (3.19) we get \(u_{kt}\rightarrow u_{t} \) a.e. in \(\varOmega \times (0,T)\). Hence
On the other hand, by the Sobolev embedding theorem and \(\|\nabla u _{kt}\|_{2}^{2}\leq L_{1}\), we have
Thus, using (3.20), (3.21), and the Lions lemma, we derive
Let \(\mathcal{D}(0,T)\) be the space of \(C^{\infty }\) functions with compact support in \((0,T)\). Multiplying (3.1) by \(\theta (t) \in \mathcal{D}(0,T)\) and integrating over \((0,T)\), it follows that
Noting that \(\{\omega_{j}\}^{\infty }_{j=1}\) is a basis of \(H_{0}^{1}\), via convergences (3.18), (3.19), and (3.22), we can get from (3.23) that
and hence, for all \(\omega \in H_{0}^{1}(\varOmega)\),
Using (3.2) and (3.19), we have
On the other hand, by Pata and Zucchi [31] we have that
Combining (3.20), (3.25), and (3.26), we complete the proof.
Remark 2.4
For the uniqueness of the weak solution, see [1].
4 Proof of Theorem 2.2
Define the functionals
and
where \(M>0 \) will be fixed later. We recall that \(E(t)\) is decreasing since \(E'(t)\leq 0\).
Lemma 4.1
For \(M>0\) sufficiently large, there exist constants \(\beta_{1},\beta_{2}>0\) such that
Proof
By the Hölder and Cauchy inequalities we have
Since \(E(t)\) is decreasing, from the Sobolev embedding theorem we have
Therefore
Then taking \(M>C\), we complete the proof. □
Lemma 4.2
There exist \(C_{2}>0\) and \(C_{3}>0\), dependent on the initial data, such that
Proof
From the definition of \(\varPsi (t)\) we get
Using (1.5), we easily see that
We now estimate the second and third terms in the right-hand side as follows. Using the Cauchy inequality with ε and the Hölder inequality, we have
and
Therefore
Noting the definitions of \(E(t) \) and \((A_{2})\), we obtain
By Sobolev embedding we have
Using (\(A_{3}\)), we get
Combining (4.3)–(4.5), we finish the proof of Lemma 4.2. □
Proof of Theorem 2.2
By Lemma 4.2 we have
Note that
and
Then taking \(M = \max ( 2C_{3},\frac{C_{2}}{\gamma } ) \), we have
Using Lemma 4.1, we obtain
By the Gronwall inequality we obtain
By Lemma 4.1 we have
that is,
where \(\nu =\frac{1}{\beta_{2}}\) and \(K=\frac{\beta_{2}E(0)}{\beta _{1}}\). □
5 Proof of Theorem 2.3
We define
and set
where M and ε will be fixed later.
Lemma 5.1
For \(M>0\) sufficiently large, there exist constants \(\beta_{1},\beta_{2}>0\) such that
for any \(0<\varepsilon \leq 1\).
Proof
Since
by the Hölder and Cauchy inequalities we have
Since \(E(t)\) is decreasing, from the Sobolev inequality we have
Therefore
Consequently,
Using Lemma 4.1 and taking M large enough, we complete the proof of Lemma 5.1. □
Lemma 5.2
Under the conditions of Theorem 2.2, the functional
satisfies
for any \(\delta_{1}>0\).
Proof
As in the proof of Lemma 4.2, we get
and
By the Cauchy inequality the second and third terms can be estimated as follows:
and
where \(\delta_{1}>0\).
Combining (5.3)–(5.6), we establish Lemma 5.2. □
Lemma 5.3
Under the conditions of Theorem 2.2, there exist constants \(C,C',C''>0\) such that
satisfies
for any \(\delta_{2}>0\).
Proof
From definition of Ψ we have
From (1.5) we see that
By the Green formula and the Cauchy and Hölder inequalities we have the following estimates:
and
Using (\(A_{2}\)) and the Cauchy, Hölder, and Poincaré inequalities, we obtain
Therefore
From (1.5) we easily obtain
Then
Using the Green formula and the Cauchy and Hölder inequalities, we have
Using the method similar to that in the proof of Lemma 4.1, we get
Then
Considering (5.7) and (5.8), we arrive at the conclusion. □
Proof of Theorem 2.3
Using
Taking \(M>0\) sufficiently large and suitable \(\varepsilon,\delta_{1}, \delta_{2}>0\) such that
by using the inequality \(\mu '(\tau)\leq -k_{1}\mu^{q}(\tau)\) we have
Therefore
By Remark 2.1 and (\(A_{2}\)), taking suitable \(M>0\), \(\varepsilon, \delta_{1}, \delta_{2}>0\), we get
Using (\(A_{3}\)), we can easily show that \(\int_{0}^{\infty } \mu^{1-\theta }(\tau)\,d\tau <\infty \) for any \(\theta <2-q\). Then
with positive constant \(L>1\).
Using the conditions of Theorem 2.3, the Hölder inequality, and (5.10), we see that
Therefore we get, for \(\sigma >1\),
By choosing \(\theta =\frac{1}{2}\) and \(\sigma =2q-1\) (hence \(\frac{\sigma \theta }{q-1+\theta }=1 \)) estimate (5.12) gives
A combination of (5.9) and (5.13) then leads to
By Lemma 5.1 we have
A simple integration of (5.14) over \((0,t)\) yields
that is,
□
6 Conclusions
In this paper, we consider the Dirichlet boundary value problem of nonlinear evolution equation with hereditary memory, variable density, and external force term. We prove the existence of a global solution by means of the Galerkin method, establish the exponential stability by using only one auxiliary functional (this method is simpler than that in [1]), and also show the polynomial stability under suitable conditions. Under suitable hypotheses on the external force term function f and integral kernel function μ with \(\gamma \geq 0\) in the model, we can further consider the local existence and blowup phenomenon of the solution.
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Acknowledgements
The authors would like to thank the referee for his/her careful reading and kind suggestions.
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FL currently works at the School of Mathematical Sciences, Qufu Normal University, P.R. China. He does research in Applied Mathematics and Analysis. He and his group are engaged in the research on the well-posedness and longtime dynamics for some nonlinear evolution equations. ZJ has got Master’s degree in Qufu Normal University and is a member of Li’s group.
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This work was supported by National Natural Science Foundation of China (No. 11201258).
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Li, F., Jia, Z. Global existence and stability of a class of nonlinear evolution equations with hereditary memory and variable density. Bound Value Probl 2019, 37 (2019). https://doi.org/10.1186/s13661-019-1152-x
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DOI: https://doi.org/10.1186/s13661-019-1152-x