Quasilinear Cauchy-Dirichlet problem for parabolic equations with $VMO_x$ coefficients

TL;DR

This paper proves the existence, uniqueness, and Holder continuity of strong solutions for a class of parabolic quasilinear equations with VMO_x coefficients and discontinuous data, advancing the understanding of such complex PDEs.

Contribution

It establishes strong solvability results for parabolic equations with coefficients of VMO_x type and discontinuous data, under new structural conditions.

Findings

Existence and uniqueness of strong solutions

Solutions are Holder continuous

Applicable to equations with VMO_x coefficients

Abstract

We study the strong solvability of the Cauchy-Dirichlet problem for parabolic quasilinear equations with discontinuous data. The principal coefficients depend on the point $(x,t)$ and on the solution u, the dependence on x is of VMO type while these are only measurable with respect to t. Assuming suitable structural conditions on the nonlinear terms, we prove existence and uniqueness of the strong solution, which turns out to be also Holder continuous.