Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity

Abstract

In this paper, the non-autonomous dynamical behavior of weakly damped wave equation with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then establish the existence of the H 1 lu (R 3) × L 2 lu (R 3), H 1 ρ (R 3) × L 2 ρ (R 3)-pullback attractor for the Shatah-Struwe solutions process of this equation. The results are based on the recent extension of Strichartz estimates for the bounded domains.

Convergence of Non-autonomous Attractors for Subquintic Weakly Damped Wave Equation

“…for which we prove the results on the existence and asymptotic smoothness of Shatah-Struwe solutions, derive the asymptotic smoothing estimates and obtain the result on the upper-semicontinuous convergence of attractors. Thus we extend and complete the previous results in [27] where only the autonomous case was considered, and in [30,31] where the nonlinearity was only in the autonomous term. We stress some key difficulties and achievements of our work.…”

“…on the three dimensional torus, where μ(t) can be a measure. Mei, Xiong, and Sun [31] obtained the existence of the pullback attractor for the problem governed by the equation u tt + u t − u + f (u) = g(t), (1.4) for the subquintic case on the space domain given by whole R 3 in the so called locally uniform spaces. Mei and Sun [30] obtained the existence of the uniform attractor for non a translation compact forcing term for the problem governed by (1.4) with subquintic f .…”

Existence of time-dependent attractor of wave equation in locally uniform space

“…To the best of our knowledge, the well-studied case is that the function ε ( t ) is a positive constant in equation (1), the existence and uniqueness of solutions together with the corresponding attractors have been established in classical locally uniform space (see Mei et al [4] and Yang et al [5]). However, when the coefficient ε ( t ) not only depends on time but also is positive decreasing function and vanish at infinity, this situation becomes much more complicated.…”

“…U(t, t − T )x − U(t, t − T )y X tρ ≤ + φ t,T (x, y) forall x, y ∈ B t ,where φ t,T (•, •) depends on t and T . Then {U(t, τ )} t≥τ is (X t , X tρ )-pullback asymptotically compact.It is easy to show the following theorem by using the ideas introduced in Mei et al[4].Theorem 2. Let {U(t, τ ) : t ≥ τ } be a process on a Banach space X t .…”

Convergence of non-autonomous attractors for subquintic weakly damped wave equation

“…on the three dimensional torus, where µ(t) can be a measure. Mei, Xiong, and Sun [31] obtain the existence of the pullback attractor for the problem governed by the equation u tt + u t − ∆u + f (u) = g(t), (1.4) for the subquintic case on the space domain given by whole R 3 in the so called locally uniform spaces. Mei and Sun [30] obtain the existence of the uniform attractor for non translation compact forcing for the problem governed by (1.4) with subquintic f .…”