Existence of weak solutions for a class of non-divergent parabolic equations with variable exponent

TL;DR

This paper proves the existence of weak solutions for a class of doubly degenerate non-divergent parabolic equations with variable growth, and shows that the support of solutions does not expand over time.

Contribution

It establishes the existence of weak solutions for different cases of the equation and demonstrates the non-expansion of the solution's support, advancing understanding of such variable exponent equations.

Findings

Existence of weak solutions for 1 ≤ m < 2 and m ≥ 2 cases.

Non-expansion of the support of solutions.

Methodological advances in handling variable growth conditions.

Abstract

A doubly degenerate parabolic equation in non-divergent form with variable growth is investigated in this paper. In suitable spaces, we prove the existence of weak solutions of the equation for cases $1\leq m < 2$ and $m\geq 2$ in different ways. And we establish the non-expansion of support of the solution for the problem.