TITLE:
The Time-Dependent Attractors for the Wave Equation with Fading Memory and Structural Damping
AUTHORS:
Xuan Wang, Luyi Yan
KEYWORDS:
Fading Memory, Structural Damping, Time-Dependent Global Attractors, Wave Equation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.2,
February
13,
2026
ABSTRACT: In this article, the asymptotic behavior of the solutions to the wave equation with fading memory and structural damping
∂
t
2
u−Δu+γ
(
−Δ
)
θ
∂
t
u−
∫
0
∞
μ(
s
)Δ
η
t
(
s
)ds
+f(
u
)=g(
x
)
is considered. First of all, when the growth exponent of nonlinear terms
f
satisfies
2≤p≤3+2θ
, the well-posedness of solutions is obtained by applying Faedo-Galerkin approximation method and energy estimation; secondly, the asymptotic compactness of the solution process is proved via the method of contraction function; finally, the existence of time-dependent global attractor is obtained in the natural energy space
H
0
1
(
Ω
)×
L
2
(
Ω
)×
L
μ
2
(
ℝ
+
;
H
0
1
(
Ω
)
)
.