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Unique and Non-Unique Fixed Points and their Applications

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Volume 2021 |Article ID 9924504 | https://doi.org/10.1155/2021/9924504

Loay Alkhalifa, Khaled Zennir, "New Estimates of Solution to Coupled System of Damped Wave Equations with Logarithmic External Forces", Journal of Function Spaces, vol. 2021, Article ID 9924504, 7 pages, 2021. https://doi.org/10.1155/2021/9924504

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

Academic Editor: Liliana Guran
Received18 Mar 2021
Revised30 Mar 2021
Accepted02 Apr 2021
Published10 Apr 2021

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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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.

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