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

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


Introduction
In the present paper, we consider an initial boundary value problem with damping terms and logarithmic sources, for x ∈ ℝ n , t > 0 where b > 0, n ≥ 3, and k is a small positive real number. The density function ρðxÞ > 0, for all x ∈ ℝ n , where ðϕðxÞÞ −1 = 1/ϕ ðxÞ ≡ ρðxÞ, under homogeneous Drichlet boundary conditions. A related initial boundary value problem was considered by Han in [1]: u tt + u t − Δu + u + u j j 2 u = u ln u j j 2 , x ∈ Ω, t ∈ 0, T ½ Þ, and the global existence of weak solutions was proved, for all ðu 0 , u 1 Þ ∈ H 1 0 × L 2 in ℝ 3 . 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 Eð0Þ < d, critical initial energy Eð0Þ = d, and the arbitrary high initial energy Eð0Þ > 0ðω = 0Þ) were proved. In [3], Al-Gharabli established explicit and general energy decay results for the problem When the density ϕðxÞ ≠ 1, 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 D 1,2 ðℝ n Þ × L 2 g ðℝ n Þ, where ðϕðxÞÞ −1 ≔ gðxÞ. 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 ℝ n , 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.

Lemma 2.
Let ρ satisfy (H3). Then, there are positive constants C S > 0 and C P > 0 that depend only on n and ρ such 2 Journal of Function Spaces for v ∈ H . 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.

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

Lemma 9.
Under the assumptions in Theorem 8, then the functional ΦðtÞ defined by satisfies for any t ≥ 0, Proof. We differentiate ΦðtÞ, using (1), we can get It follows from Young and Poincaré's inequality that for any ε > 0, Exploit Young and Poincaré's inequalities to estimate Inserting (32)-(33) into (31) yields for any ε > 0, Taking ε > 0 small enough in (34) such that The proof is hence complete.
Lemma 10. Under the assumptions in Theorem 8, then the functional ψðtÞ defined by Journal of Function Spaces satisfies for any δ > 0, Proof. Taking the derivative of ψðtÞ and using (1), we conclude that We then use Young and Poincaré's inequalities; we can get for any δ > 0, 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 ε 0 ∈ ð0, 1Þ and gðsÞ = s ε 0 ðjln sj − sÞ. Notice that g is continous on ð0, ∞Þ, its limit at 0 is 0, and its limit at ∞ is −∞. Then, g has a maximum m ε 0 on ½0, ∞Þ, so the following inequality holds s ln s j j≤ s 2 + m ε 0 s 1−ε 0 , for all s > 0: ð43Þ Using the Cauchy-Schwartz's inequality and applying (43), yields, for any δ > 0, Therefore, the proof is complete. Now, we define a Lyapunov functional LðtÞ by where M, ε 1 , and ε 2 are positive constants which will be taken later.
It is easy to see that LðtÞ and EðtÞ are equivalent in the sense that there exist two positive constants β 1 and β 2 such that Remark 11 (see [3]). Since ζ i is nonincreasing, we have Proof of Theorem 8. For any fixed t 0 > 0, we have for any t ≥ t 0 , It follows from (37), (30), and (20) that