An optimal boundedness result for weak solutions of double phase quasilinear parabolic equations

TL;DR

This paper proves local boundedness of weak solutions for a class of double phase quasilinear parabolic equations, extending understanding of solution regularity under specific growth and boundedness conditions.

Contribution

It establishes an optimal boundedness result for weak solutions of double phase parabolic equations with new restrictions on parameters and coefficient functions.

Findings

Weak solutions are locally bounded under given conditions.

The result applies to equations with measurable coefficients and specific growth restrictions.

Provides a foundation for further regularity analysis of double phase equations.

Abstract

We obtain local boundedness of weak solutions of double phase quasilinear parabolic equations of the form \[u_t-\text{div} \left(|\nabla u|^{p-2}\nabla u+a(x,t)|\nabla u|^{q-2}\nabla u\right)=0,\] where, we have imposed the restrictions $\frac{2N}{N+2}<p<\infty$, $0\leq a(x,t)\leq M$ is measurable and $q < p\frac{N+1}{N-1}$.