The mixed Cauchy-Dirichlet problem for a viscous Hamilton-Jacobi equation

Abstract

We study the existence, uniqueness, and regularity of weak solutions for a viscous Hamilton-Jacobi equation of the form: $u_t-\Delta u=a|\nabla u|^p, $ $p\in(0,\infty)$ and $a\in{{\bf R}}$, $a\neq 0$, with Dirichlet boundary condition and irregular initial data $\mu_0$. The cases of initial data $\mu_0$ a bounded Radon measure, or a function in the Lebesgue space $L^q, 1\leq q < \infty$ are investigated.