Abstract
A nonlinear wave equation of Kirchhoff type with memory condition at the boundary in a bounded domain is considered. We establish a general decay result which includes the usual exponential and polynomial decay rates. Furthermore, our results allow certain relaxation functions which are not necessarily of exponential and polynomial decay. This improves earlier results in the literature. MSC: 35L05; 35L70; 35L75; 74D10.
Highlights
1 Introduction In this article, we study the asymptotic behavior of the energy function related to a nonlinear wave equation of Kirchhoff type subject to memory condition at the boundary as follows: utt − M ||∇u||22 u + l(t)h(∇u) − ut + a(x)f (u) = 0 in × (0, ∞), (1:1)
3 Decay of solutions we study the asymptotic behavior of the solutions of system (1.1)-(1.4) when the resolvent kernel k satisfies k(0) > 0, k(t) ≥ 0, k (t) ≤ 0, k (t) ≥ −γ (t)k (t), (3:1)
Messaoudi and Soufyane in 2010 [17] considered a semi-linear wave equation, in a bounded domain, where the memory-type damping is acting on the boundary
Summary
We study the asymptotic behavior of the energy function related to a nonlinear wave equation of Kirchhoff type subject to memory condition at the boundary as follows: utt − M ||∇u||22 u + l(t)h(∇u) − ut + a(x)f (u) = 0 in × (0, ∞),. Messaoudi and Soufyane [17] studied the following problem: utt − u + f (u) = 0 in a bounded domain with boundary conditions (1.7)-(1.9) They improved the results of [15] by applying the multiplier techniques. They proved that the energy decays with the same rate of decay of the relaxation function This latter result improved an earlier one by Park et al [23], where the authors considered (1.10) in a bounded domain with nonlinear boundary damping and memory term and M(s) = 1 + s and f = 0.
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