Potential estimates and quasilinear parabolic equations with measure data

Abstract

In this paper, we study the existence and regularity of the quasilinear parabolic equations: $$u_t-\mathrm{div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu ,$$ 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 $N\ge 2$. 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 $q>1$

110 pages. Some typos corrected, to appear in the Memoirs of the AMS

Cited by in corpus (1)

• An elliptic semilinear equation with source term and boundary measure data: the supercritical case