Asymptotics of singularly perturbed damped wave equations with super-cubic exponent

Abstract

This work is devoted to studying the relations between the asymptotic behavior for a class of hyperbolic equations with super-cubic nonlinearity and a class of heat equations, where the problem is considered in a smooth bounded three dimensional domain. Based on the extension of the Strichartz estimates to the case of bounded domain, we show the regularity of the pullback, uniform, and cocycle attractors for the non-autonomous dynamical system given by hyperbolic equation. Then we prove that all types of non-autonomous attractors converge, upper semicontiously, to the natural extension global attractor of the limit parabolic equations.