Attractors for parabolic problems with p(x)-Laplacian: Bounds, continuity of the flow and robustness

Abstract

In this work we consider a family of quasilinear equations with variable exponents ($p\left(x\right)$-Laplacian) and perturbations which are not globally Lipschitz. We prove existence of global solutions, existence of global attractors and we provide conditions on the data in order that the associated semilinear equation ($p\left(x\right)\equiv 2$) commands the asymptotic dynamics of the family of problems when the exponents are sufficiently close to 2 (uniformly in x) by showing the continuity of the flows and the upper semicontinuity of the global attractors.