Attractors for non-autonomous wave equations with a new class of external forces

Shan Ma, Xiyou Cheng

Abstract:

We introduce a new concept Condition (C*), and denote the set of all functions satisfying
Condition (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 we show that there are many functions satisfying Condition (C*); then,
in application, we obtain the existence of uniform attractors in $E_0 = H_0^1 \times L^2$ for non-
autonomous wave equations involving mixed differential terms with this new
class of time dependent external forces $h(x,t) \in L_{n\text{ex}}^2(\mathbb{R};X)$.