Potential estimates and quasilinear parabolic equations with measure data

TL;DR

This paper investigates the existence, regularity, and gradient estimates for solutions to quasilinear parabolic equations with measure data, introducing new results for nonlinearities involving the solution and its gradient.

Contribution

It establishes existence results for measure data problems with specific nonlinearities and derives new gradient estimates under minimal boundary and nonlinearity conditions.

Findings

Existence of solutions with nonlinearities involving the solution for q>1.

Global weighted-Lorentz, Lorentz-Morrey, and Capacitary estimates on gradients.

Solutions exist for equations with gradient nonlinearities | abla u|^q for q>1.

Abstract

In this paper, we study the existence and regularity of the quasilinear parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu,$$ in either $\mathbb{R}^{N+1}$ or $\mathbb{R}^N\times(0,\infty)$ or on a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$ where $N\geq 2$. In this paper, we shall assume that the nonlinearity $A$ fulfills standard growth conditions, the function $B$ is a continuous and $\mu$ is a radon measure. Our first task is to establish the existence results with $B(u,\nabla u)=\pm|u|^{q-1}u$, for $q>1$. We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with $B\equiv 0$, under minimal conditions on the boundary of domain and on nonlinearity $A$. Finally, due to these estimates, we solve the existence problems with $B(u,\nabla u)=|\nabla u|^q$ for $q>1$