Journal of Function Spaces
Volume 2021 (2021), Article ID 9924504, 7 pages
https://doi.org/10.1155/2021/9924504

Research Article

New Estimates of Solution to Coupled System of Damped Wave Equations with Logarithmic External Forces

Loay Alkhalifa¹ and Khaled Zennir1,2

¹Department of Mathematics, College of Sciences and Arts, Ar-Rass, Qassim University, Saudi Arabia
²Laboratoire de Mathématiques Appliquées et de Modélisation, Université 8 Mai 1945 Guelma, B.P. 401 Guelma 24000, Algeria

Correspondence should be addressed to Khaled Zennir; k.zennir@qu.edu.sa

Received 18 March 2021; Revised 30 March 2021; Accepted 2 April 2021; Published 10 April 2021

Academic Editor: Liliana Guran

Copyright © 2021 Loay Alkhalifa and Khaled Zennir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In the paper, we consider new stability results of solution to class of coupled damped wave equations with logarithmic sources in . We prove a new scenario of stability estimates by introducing a suitable Lyapunov functional combined with some estimates.

1. Introduction

In the present paper, we consider an initial boundary value problem with damping terms and logarithmic sources, for

where , , and is a small positive real number. The density function , for all , where , under homogeneous Drichlet boundary conditions.

A related initial boundary value problem was considered by Han in [1]:

and the global existence of weak solutions was proved, for all in . The weak and strong damping terms in logarithmic wave equation

were introduced by Lian and Xu [2]. The global existence, asymptotic behavior, and blowup at three different initial energy levels (subcritical energy , critical initial energy , and the arbitrary high initial energy ) were proved. In [3], Al-Gharabli established explicit and general energy decay results for the problem

When the density , Papadopoulos and Stavrakakis [4] considered the following semilinear hyperbolic initial value problem:

The authors proved local existence of solutions and established the existence of a global attractor in the energy space , where . Miyasita and Zennir [5] proved the global existence of the following viscoelastic wave equation:

The novelty of our work lies primarily in the use of a new condition between the weights of damping the external forces, where we outline the effects of the damping term with less conditions on the viscoelastic terms. We also propose logarithmic nonlinearities in sources and used classical arguments to estimate them. These nonlinearities make the problem very interesting in the application point of view. In order to compensate for the lack of classical Poincaré’s inequality in , we use the weighted function to use generalized Poincaré’s one. The main contribution of this paper is introduced in Theorem 8, where we obtain decay estimates with positive initial energy under a general assumption on the kernel. The rest of the paper is outline as follows. In Section 2, we give some preliminaries and our main results. In Section 3, we will prove the general decay of energy to the problem.

2. Preliminaries and Main Results

We state some assumptions and definitions that will be useful in this paper. With respect to the relaxation functions , we assume for

(H1) satisfy for any ,

(H2) There exist nonincreasing differentiable functions that satisfy

(H3) The function with and , where

Definition 1 (see [4]). We define the function spaces of our problem and their norms as follows:

Let the function spaces as the closure of with respect to the norm for the inner product:

and be defined with the norm for

For general , is the weighted space under a weighted norm

To distinguish the usual space from the weighted one, we denote the standard norm by

We denote an eigenpair by

for any . Then, according to [4],

holds and is a complete orthonormal system in .

Now, we introduce Sobolev embedding and generalized Poincaré’s inequalities.

Lemma 2. Let satisfy (H3). Then, there are positive constants and that depend only on and such that

for .

Lemma 3 (see Lemma 2.2 in [6]). Let satisfy (H3). Then, we have

for , where for .

The energy functional associated to problem (1) is given by

where

With direct differentiation of (18), using (1), we obtain

which let our system dissipative.

Lemma 4 (see [7]) (logarithmic Sobolev inequality). Lets be any function in and be any number. Then,

Lemma 5 (see [8]) (logarithmic Gronwall inequality). Let , and assume that the function satisfies

then

We define the following functionals

Then, we introduce

Lemma 6. Let such that and . Then, we have

Theorem 7 (see [5]). Let . Under the assumptions (H1)–(H3). Then, problem (1) has a global weak solution in the space

Then, the main result in this paper is the general decay of energy to problem (1) which is given in the following theorem.

Theorem 8. Assume the assumptions (H1)–(H3) hold and . Let be the weak solution of problem (1) with the initial data . Then, there exist constant such that the energy defined by (18) satisfies for all ,

3. Asymptotic Behavior for

The following technical lemmas are useful to prove the general decay of energy to problem (1).

Lemma 9. Under the assumptions in Theorem 8, then the functional defined by

satisfies for any ,

Proof. We differentiate , using (1), we can get

It follows from Young and Poincaré’s inequality that for any ,

Exploit Young and Poincaré’s inequalities to estimate

Inserting (32)–(33) into (31) yields for any ,

Taking small enough in (34) such that

The proof is hence complete.

Lemma 10. Under the assumptions in Theorem 8, then the functional defined by

satisfies for any ,

Proof. Taking the derivative of and using (1), we conclude that

We then use Young and Poincaré’s inequalities; we can get for any ,

The second and third terms can be treated as

The fourth and fifth terms will be estimated by

respectively.
For the last term, we have

Let and . Notice that is continous on , its limit at is , and its limit at is . Then, has a maximum on , so the following inequality holds

Using the Cauchy-Schwartz’s inequality and applying (43), yields, for any ,

Combining (39)–(44) with (39) gives us (37) with

Therefore, the proof is complete.

Now, we define a Lyapunov functional by

where , , and are positive constants which will be taken later.

It is easy to see that and are equivalent in the sense that there exist two positive constants and such that

Remark 11 (see [3]). Since is nonincreasing, we have

Proof of Theorem 8. For any fixed , we have for any ,

It follows from (37), (30), and (20) that

Using the logarithmic Sobolev inequality, we have

Recalling (18) and , we get

Now, we take small enough so that

For any fixed , we pick so small that

On the other hand, we choose large enough so that (47) holds, and further

We can conclude that there exist two positive constant and such that

Multiplying (56) by by (H2) and use the fact that

and (48), we get

Multiply (58) by and recall that to obtain

Using Young’s inequality, for any ,

which implies

It is clear that to get

By using (61) and , we arrive at

Integration over leads to for some constant such that

The equivalence of and completes Proof of Theorem 8.

Remark 12.

(1)We mention here that we have coupled our system without the classical way, i.e., our idea is not to couple equations in the logarithmic nonlinear terms(2)Most contribution here is to obtain our nonexistence result with less conditions on the viscoelastic terms

Data Availability

No data were used in this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

References

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