Global a priori bounds for weak solutions to quasilinear parabolic equations with nonstandard growth

TL;DR

This paper establishes global a priori bounds for weak solutions to a broad class of quasilinear parabolic equations with nonstandard growth, including variable exponent cases, using localization and De Giorgi's iteration techniques.

Contribution

It introduces new global bounds for weak solutions to quasilinear parabolic problems with nonstandard growth, extending previous results even for constant exponent cases.

Findings

Derived global a priori bounds for weak solutions

Applicable to equations with variable exponent p(t,x)

Results are new even for constant exponent cases

Abstract

In this paper we study a rather wide class of quasilinear parabolic problems with nonlinear boundary condition and nonstandard growth terms. It includes the important case of equations with a $p(t,x)$-Laplacian. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems. Our results seem to be new even in the constant exponent case.