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Weighted integral inequality and applications in general energy decay estimate for a variable density wave equation with memory
Boundary Value Problems volume 2018, Article number: 164 (2018)
Abstract
This paper develops a weighted integral inequality to derive decay estimates for the quasilinear viscoelastic wave equation with variable density
with initial conditions and boundary condition, where g is a memory kernel function and ρ is a positive constant. Depending on the properties of convolution kernel g at infinity, we establish a general decay rate of the solution such that the exponential and polynomial decay results in some literature are special cases of this paper, and we improve the integral method used in the literature.
1 Introduction
Using Lyapunov technique for some perturbed energy, Messaoudiet and Khulaifi [24] studied the following problem:
where \(\varOmega \subset \mathbb{R}^{n}\) (\(n\geq 1\)) is a bounded domain with smooth boundary ∂Ω such that the divergence theorem can be applied.
Cavalcanti et al. [3] proved that the finite energy solutions of nonlinear abstract PDE with a memory term exhibit exponential decay rates when strong damping \(-\Delta u_{t}\) is active, this uniform decay is no longer valid (by spectral analysis arguments) for dynamics subjected to frictional damping only, say \({g=0}\). Viscoelastic equations with variable density have been studied by many authors, and several stability results have been established in [2, 9, 10, 23, 25, 26]. As we know, all the stability results were obtained by establishing differential inequalities on the functional equivalent to the original energy. Our approach is based on integral inequalities and multiplier techniques. Indeed, instead of using Lyapunov technique for some perturbed energy, we rather concentrate on the original energy, showing that it satisfies a weighted integral inequality which, in turn, yields the final decay estimate. We prove a general decay rate from which the exponential decay and the polynomial decay are only special cases. Due to the assumption on g, the weighted inequality established in this paper improves the integral inequality in [1].
We mention here some related works concerning the energy for the evolution equations. For the nonlinear damped wave equations and Marguerre–von Karman system, some energy decay rate estimates were obtained in [11–15, 19, 20] and the references therein. Li and his coauthors [7, 16–18] studied the blow up phenomena of the solutions for evolution equations. This research laid a good foundation for our further study. For the stability and convergence results of evolution equations, the readers can refer to [6, 21, 22].
The outline of this paper is as follows. In Sect. 2, we present the preliminaries and our important result. In Sect. 3, we construct an energy inequality, prove the main Theorem 2.2 and give applications to various functions \(\xi (t)\).
To simplify calculations in our analysis, we introduce the following notations:
2 Preliminaries and main result
In this section we prepare some material needed in the proof of our result and state our main result. Throughout this paper, C denotes a generic positive constant. We impose the following assumptions on ρ and g.
Assumption 2.1
Set \(G(t) =\int_{0}^{t} g(s)\,ds\). We assume that
-
1.
$$\begin{aligned} 0< \rho \leqslant \frac{2}{n-2} \quad \text{if } n\geq 3; \quad\quad \rho >0 \quad \text{if } n=1,2, \end{aligned}$$
which implies that
$$\begin{aligned} H_{0}^{1}(\varOmega )\hookrightarrow L^{2(\rho +1)}( \varOmega ). \end{aligned}$$ -
2.
\(g: [0, \infty ) \rightarrow [0, \infty )\) is a locally absolutely continuous function such that
$$\begin{aligned} G(\infty ) < 1,\quad\quad g(0) > 0, \quad\quad g'(t) \leqslant 0, \quad \text{for a.e. } t \geqslant 0. \end{aligned}$$ -
3.
There exists a non-increasing function \(\xi \in C^{1}[0,+\infty )\) with \(\int_{0}^{+\infty }\xi (\tau )\,d\tau =+\infty \) satisfying
$$\begin{aligned} g'(t)\leqslant -\xi (t)g(t),\quad\quad \xi (t)>0, \quad \forall t \geqslant 0. \end{aligned}$$
Theorem 2.1
([24])
Suppose that \(( u_{0},u _{1} ) \in H^{1}_{0}(\varOmega )\times H^{1}_{0}(\varOmega )\). Under Assumption 2.1, there exists a unique solution u of (1.1) satisfying
and
Lemma 2.1
\(H_{0}^{1}(\varOmega )\hookrightarrow L^{r}(\varOmega )\) with
which implies
Lemma 2.2
Let u be the global solution of the problem (1.1), then for any suitable function w one has
Proof
From
we have
□
Lemma 2.3
Let u be the global solution of the problem (1.1), then
where
Proof
Multiplying equation (1.1) by \(u_{t}\), integrating by parts over Ω and using Lemma 2.2, we obtain the conclusion. □
Our main result is the following decay theorem.
Theorem 2.2
Let u be the global solution of problem (1.1) with Assumption 2.1. We define the energy functional as
Then, for some \(t_{0}>0\) there exist positive constants \(C_{0}\) and ω such that
3 The proof of main result
In order to derive the desired result of Theorem 2.2 by the integral method, we establish the following weighted integral inequality.
Lemma 3.1
Let u be the solution of (1.1) under Assumption 2.1, then
for some constant \(C>0\).
To prove the above inequality, we need the following two lemmas.
Lemma 3.2
Let u be the solution of (1.1) under Assumption 2.1, then
Proof
Multiplying by \(\xi (t)u(t)\) both sides of equation (1.1), integrating the resulting equation over \(\varOmega \times [S,T]\) (\(0\leqslant S\leqslant T\)), then using the boundary conditions and Lemma 2.2, we have
According to the definition of the energy functional \(E(t)\), we get
Combining (3.1) with (3.2), we deduce that
From Lemma 2.3, we see that
that is,
which, together with Assumption 2.1, implies
For the fourth term on the right-hand side of (3.3), integrating by parts and using Lemma 2.2, we have
By Young inequality, Lemma 2.1 and the definition of \(E(t)\), we have
with some positive constant \(k_{1}\). Hence,
Similarly,
For the fifth term on the right-hand side of (3.3), integrating by parts, we have
with some positive constant \(k_{2}\).
For the sixth term on the right-hand side of (3.3), we have
Combining with (3.4), we obtain
Integrating by parts (and noting \(E'(t)\leqslant 0\)), we have
and
Owing to (3.10)–(3.12), we get
Choosing ε small enough, we obtain from (3.13) that
The proof of Lemma 3.2 is completed. □
Lemma 3.3
Let u be the solution of (1.1) under Assumption 2.1, then
Proof
Multiplying by \(\xi (t)\int_{0}^{t}g(t-s) ( u(s)-u(t) ) \,ds\) both sides of equation (1.1) and then integrating the resulting equation over \(\varOmega \times [S,T]\) (\(0\leqslant S\leqslant T\)) gives
Integrating by parts and using Lemma 2.2, we obtain
Moreover, we have
and
Therefore, plugging the above three identities into (3.14), we get
Using Cauchy and Hölder inequalities as well as Lemma 2.1, we have
which implies that
In order to estimate the second term on the right-hand side of (3.15), we apply (3.16) and (3.11) to get
In addition, for any \(\delta >0\), using Young inequality, we have
Using Hölder inequality and Lemmas 2.1 and 2.3, we have
Therefore,
Since
one obtains
Using (3.20) and (3.11), we have
Similarly,
Now, since \(g(t)\geqslant 0\) and \(G(\infty )<1\), we get
and
Combining (3.15)–(3.25), we obtain
Since g is continuous and \(g(0)>0\), for any \(t_{0}>0\), we have
Now, if we fix \(\delta >0\) small enough such that
then by (3.26), for \(T>S\geqslant t_{0}\), we have
□
Proof of Lemma 3.1
Plugging the estimate of Lemma 3.3 into the inequality of Lemma 3.2, and fixing ε small enough, we obtain
for some constant \(C>0\). Letting \(T\rightarrow +\infty \), we have
with
□
Proof of Theorem 2.2
From Assumption 2.1 and Lemma 2.3, we know that \(E(t)\) is a non-increasing function and \(\psi :[t_{0},+ \infty )\rightarrow \mathbb{R}^{+}\) is a strictly increasing \(C^{2}\) function such that \(\psi (t_{0})=0\) and \(\lim_{t\rightarrow +\infty } \psi (t)=+\infty \). Firstly, we define a new function \(f:[t_{0},\infty )\rightarrow \mathbb{R}^{+}\) as follows:
then f is a non-increasing function such that
Set \(t=\psi (S)\). Since \(\lim_{T\rightarrow +\infty }\psi (T)=+ \infty \), we get
Next, we define the following function:
where \(c>0\) is a constant. Noting (3.28), we obtain
Integrating (3.30) over \([t_{0},t]\) and noting (3.28), we have
Furthermore, using (3.29) and (3.31), we obtain
On the other hand, noting that f is a positive non-increasing function, it is easy to see that
Combining (3.32) with (3.33), we get
Letting \(s=t+c\) in (3.34), we have
that is,
Moreover, letting \(t=\psi^{-1}(s)\) in (3.35), we get
that is,
for some constants \(C_{0}\) and \(\omega =\frac{1}{c}>0\).
The proof of Theorem 2.2 is completed. □
Remark 3.1
From Theorem 2.2, if we choose different \(\xi (t)\), we can get different decay results. Choosing \(\xi (t) \equiv {a}\), we get the exponential decay result
Now consider \(\xi (t)=\frac{1}{(1+t)^{\gamma }}\) (\(0<\gamma \leqslant 1\)). If \(\gamma =1\), we get the polynomial decay result
If \(0<\gamma <1\), we get a decay result of the form
4 Conclusions
In this paper, we present a weighted integral inequality to derive decay estimates for a quasilinear viscoelastic wave equation with variable density and memory. Due to the assumption on the memory kernel function, the weighted inequality established in this paper improves the integral inequality in [1]. We establish a general decay rate of the solution such that the exponential and polynomial decay results are special cases of this paper.
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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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Fushan Li 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. Fengying Hu is studying for a Master’s degree in Qufu Normal University. She 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., Hu, F. Weighted integral inequality and applications in general energy decay estimate for a variable density wave equation with memory. Bound Value Probl 2018, 164 (2018). https://doi.org/10.1186/s13661-018-1085-9
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DOI: https://doi.org/10.1186/s13661-018-1085-9