Upper-semicontinuity of uniform attractors for the non-autonomous viscoelastic Kirchhoff plate equation with memory
Yuming Qin, Hongli Wang
TL;DR
The paper analyzes the non-autonomous viscoelastic Kirchhoff plate equation with memory to understand its long-time dynamics. By developing a memory-augmented phase space and employing energy methods, it proves the existence of weak solutions, constructs a uniformly absorbing set, and establishes uniform asymptotic compactness to obtain a compact uniform attractor for subcritical and critical nonlinearities. It then shows upper semicontinuity of the uniform attractors as the damping perturbation ε→0, ensuring stability of the long-term dynamics under small damping changes. Collectively, the work extends uniform attractor theory to non-autonomous, memory-influenced plate equations and resolves questions about the limiting behavior of attractors with memory and critical nonlinearities.
Abstract
This paper delves into the long-time dynamics of a non-autonomous viscoelastic Kirchhoff plate equation with memory effects, described by $$ u_{t t}-Δu_{t t}+a_ε(t) u_t+αΔ^2 u-\int_0^{\infty} μ(s) Δ^2 u(t-s) \mathrm{d} s-Δu_t+f(u)=g(x,t), $$ in bounded domain $Ω\subset \mathbb{R}^N$ with smooth boundary and nonlinear terms. Initially, the global existence of a weak solution that induces a continuous process is established. Subsequently, the existence of a uniform attractor is demonstrated in both subcritical and critical growth scenarios, utilizing operator techniques and an innovative analytical approach. Finally, the upper semicontinuity of the family of uniform attractors as the pert parameterurbation $ε\to 0^+$ is proven through delicate energy estimates and a contradiction argument. Our results not only extend classical attractor theory to more general non-autonomous viscoelastic systems but also resolve open questions regarding the limiting behavior of attractors in the presence of both memory and critical nonlinearity.