Global attractors in stronger norms for a class of parabolic systems with nonlinear boundary conditions

TL;DR

This paper establishes the existence of global attractors in high-regularity norms for certain quasilinear parabolic systems with nonlinear boundary conditions, using maximal regularity and weighted estimates.

Contribution

It introduces a novel approach combining maximal $L_p$-regularity with temporal weights to prove attractor existence in high-regularity spaces for parabolic systems.

Findings

Existence of a compact local solution semiflow in high-regularity phase space.

Global attractors exist under a priori estimates in lower norms.

Regularity improvement enhances convergence to attractors.

Abstract

For a class of quasilinear parabolic systems with nonlinear Robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity. We further show that a priori estimates in lower norms are sufficient for the existence of a global attractor in this phase space. The approach relies on maximal $L_p$-regularity with temporal weights for the linearized problem. An inherent smoothing effect due to the weights is employed for gradient estimates. In several applications we can improve the convergence to an attractor by one regularity level.