Journal of Function Spaces
Volume 2021, Issue 1 9924504
Research Article
Open Access

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

Loay Alkhalifa, 

Loay Alkhalifa

Department of Mathematics, College of Sciences and Arts, Ar-Rass, Qassim University, Saudi Arabia qu.edu.sa

Search for more papers by this author
Khaled Zennir, 

Corresponding Author

Khaled Zennir

Department of Mathematics, College of Sciences and Arts, Ar-Rass, Qassim University, Saudi Arabia qu.edu.sa

Laboratoire de Mathématiques Appliquées et de Modélisation, Université 8 Mai 1945 Guelma, B.P. 401 Guelma 24000, Algeria univ-guelma.dz

Search for more papers by this author
First published: 10 April 2021
https://doi.org/10.1155/2021/9924504
Academic Editor: Liliana Guran

Abstract

In the paper, we consider new stability results of solution to class of coupled damped wave equations with logarithmic sources in ℝn. 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 x ∈ ℝn, t > 0
(1)
where b > 0, n ≥ 3, and k is a small positive real number. The density function ρ(x) > 0, for all x ∈ ℝn, where (ϕ(x))−1 = 1/ϕ(x) ≡ ρ(x), under homogeneous Drichlet boundary conditions.
A related initial boundary value problem was considered by Han in [1]:
(2)
and the global existence of weak solutions was proved, for all in ℝ3. The weak and strong damping terms in logarithmic wave equation
(3)
were introduced by Lian and Xu [2]. The global existence, asymptotic behavior, and blowup at three different initial energy levels (subcritical energy E(0) < d, critical initial energy E(0) = d, and the arbitrary high initial energy E(0) > 0(ω = 0)) were proved. In [3], Al-Gharabli established explicit and general energy decay results for the problem
(4)
When the density ϕ(x) ≠ 1, Papadopoulos and Stavrakakis [4] considered the following semilinear hyperbolic initial value problem:
(5)
The authors proved local existence of solutions and established the existence of a global attractor in the energy space , where (ϕ(x))−1≔g(x). Miyasita and Zennir [5] proved the global existence of the following viscoelastic wave equation:
(6)

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 ℝn, 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 ϖ1, ϖ2, we assume for i = 1, 2.

(H1) ϖ1, ϖ2 ∈ C1(ℝ+, ℝ+) satisfy for any t ≥ 0,
(7)
(H2) There exist nonincreasing differentiable functions ζ1, ζ2 : ℝ+⟶ℝ+ that satisfy
(8)

(H3) The function with γ ∈ (0, 1) and ρ ∈ Ls(ℝn)∩L∞(ℝn), where s = 2n/2n − qn + 2q

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

(9)

Let the function spaces as the closure of with respect to the norm for the inner product:
(10)
and be defined with the norm for
(11)
For general q ∈ [1, +∞), is the weighted Lq space under a weighted norm
(12)
To distinguish the usual Lq space from the weighted one, we denote the standard Lq norm by
(13)
We denote an eigenpair by
(14)
for any j ∈ ℕ. Then, according to [4],
(15)
holds and {wj} is a complete orthonormal system in .

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

Lemma 2. Let ρ satisfy (H3). Then, there are positive constants CS > 0 and CP > 0 that depend only on n and ρ such that

(16)
for .

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

(17)
for , where s = 2n/(2n − qn + 2q) for 1 ≤ q ≤ 2n/(n − 2).

The energy functional associated to problem (1) is given by
(18)
where
(19)
With direct differentiation of (18), using (1), we obtain
(20)
which let our system dissipative.

Lemma 4 (see [7] logarithmic Sobolev inequality.)Lets u be any function in and a > 0 be any number. Then,

(21)

Lemma 5 (see [8] logarithmic Gronwall inequality.)Let c > 0, γ ∈ L1(0, T; ℝ+), and assume that the function ω : [0, T]⟶[1, ∞) satisfies

(22)
then
(23)

We define the following functionals
(24)
Then, we introduce
(25)

Lemma 6. Let such that and I(t0) > 0. Then, we have

(26)

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

(27)

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 (v1, v2) be the weak solution of problem (1) with the initial data . Then, there exist constant β > 0 such that the energy defined by (18) satisfies for all t > 0,

(28)

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 Φ(t) defined by

(29)
satisfies for any t ≥ 0,
(30)

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

(31)

It follows from Young and Poincaré’s inequality that for any ε > 0,

(32)

Exploit Young and Poincaré’s inequalities to estimate

(33)

Inserting (32)–(33) into (31) yields for any ε > 0,

(34)

Taking ε > 0 small enough in (34) such that

(35)

The proof is hence complete.

Lemma 10. Under the assumptions in Theorem 8, then the functional ψ(t) defined by

(36)
satisfies for any δ > 0,
(37)

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

(38)

We then use Young and Poincaré’s inequalities; we can get for any δ > 0,

(39)

The second and third terms can be treated as

(40)

The fourth and fifth terms will be estimated by

(41)
respectively.

For the last term, we have

(42)

Let ε0 ∈ (0, 1) and . Notice that g is continous on (0, ∞), its limit at 0 is 0, and its limit at ∞ is −∞. Then, g has a maximum on [0, ∞), so the following inequality holds

(43)

Using the Cauchy-Schwartz’s inequality and applying (43), yields, for any δ > 0,

(44)

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

(45)

Therefore, the proof is complete.

Now, we define a Lyapunov functional by
(46)
where M, ε1, and ε2 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 β1 and β2 such that
(47)

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

(48)

Proof of Theorem 8. For any fixed t0 > 0, we have for any t ≥ t0,

(49)

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

(50)

Using the logarithmic Sobolev inequality, we have

(51)

Recalling (18) and , we get

(52)

Now, we take ε1 > 0 small enough so that

(53)

For any fixed ε1 > 0, we pick δ > 0 so small that

(54)

On the other hand, we choose M > 0 large enough so that (47) holds, and further

(55)

We can conclude that there exist two positive constant m and C′ such that

(56)

Multiplying (56) by ζ(t) = min{ζ1, ζ2} by (H2) and use the fact that

(57)
and (48), we get
(58)

Multiply (58) by and recall that ζ′(t) ≤ 0 to obtain

(59)

Using Young’s inequality, for any δ > 0,

(60)
which implies
(61)

It is clear that to get

(62)

By using (61) and ζ′(t) ≤ 0, we arrive at

(63)

Integration over (t0, t) leads to for some constant m′ > 0 such that

(64)

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

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.

Data Availability

No data were used in this study.

    The full text of this article hosted at iucr.org is unavailable due to technical difficulties.