TITLE:
The Existence of Global Attractor for Kirchhoff-Type Strongly Damped Wave Equation with Nonlinear Memory
AUTHORS:
Jixin Xu, Xinghui Wang
KEYWORDS:
Wave Equation, Nonlinear Memory, Global Attractor
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.4,
April
15,
2025
ABSTRACT: This paper addresses the existence of a global attractor for Kirchhoff-type strongly damped wave equation with nonlinear memory effects. The key innovation of our work lies in reformulating the historical memory term as a convolution integral involving the memory kernel
μ
and a nonlinear power-law function
|
u(
s
) |
β
u(
s
)
. First, we rigorously establish the existence, uniqueness, and regularity of solutions for Equation (1.2) through a systematic application of a priori energy estimates and the Faedo-Galerkin approximation method, while simultaneously demonstrating the presence of a bounded absorbing set. To analyze the asymptotic dynamics, we decompose the solution semigroup
S(
t
)
into two components:
S
1
(
t
)
, governed by higher-order regularity, and
S
2
(
t
)
, capturing dissipative effects. The compactness of
S
1
(
t
)
is established via operator regularity analysis combined with the compact sobolev embedding theorem, while the uniform exponential decay of
S
2
(
t
)
is proven through refined energy estimation techniques. By synthesizing these results, we conclusively demonstrate the existence of a global attractor for the system under study, thereby extending the theoretical framework for nonlinear wave equations with memory-driven dissipation.