Local boundedness for solutions to degenerate parabolic double phase problems

TL;DR

This paper establishes local boundedness of solutions to a class of degenerate parabolic double phase equations, using Caccioppoli inequalities and embedding theorems within an iterative framework.

Contribution

It introduces new techniques for proving local boundedness of weak solutions to degenerate double phase parabolic equations.

Findings

Proved local boundedness of weak solutions.

Derived Caccioppoli inequality for the double phase problem.

Established a parabolic embedding theorem applicable to the problem.

Abstract

In this paper, we investigate the local boundedness of weak solutions to degenerate parabolic double phase equation of type $$ u_t-\textrm{div}(|Du|^{p-2}Du+a(x,t)|Du|^{q-2}Du)=0\quad \text{in } \Omega_T := \Omega\times (0,T), $$ where $0\leq a(\cdot)\in L^\infty(\Omega_T)$. To this end, we derive the Caccioppoli inequality and a parabolic embedding theorem, which are then utilized in an iteration method.