The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces

Shan Ma

Abstract:

For weakly damped non-autonomous hyperbolic equations, we introduce a new concept Conditions (C*), denote
the set of all functions satisfying Conditions (C*) by $L_{n\text{ex}}^2(\mathbb{R};X)$ which are translation bounded but not translation
compact in $L_{\text{loc}}^2(\mathbb{R};X)$, and show that there are many functions satisfying Condition (C*); then we study the
uniform attractors for weakly damped non-autonomous hyperbolic equations with this new class of time
dependent external forces $g(x,t)$ is an element of $L_{n\text{ex}}^2(\mathbb{R};X)$ and prove the existence of the uniform attractors for
the family of processes corresponding to the equations in $H_0^1(\Omega)\times L^2(\Omega)$ and $D(A)\times H_0^1(\Omega)$.