Initial trace of solutions of Hamilton-Jacobi parabolic equation with absorption

Abstract

Here we study the initial trace problem for the nonnegative solutions of the equation \[ u\_{t}-Δu+|\nabla u|^{q}=0 \] in $Q\_{Ω,T}=Ω\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $Ω=\mathbb{R}^{N},$ or $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ and $u=0$ on $\partialΩ\times\left( 0,T\right) .$ We can define the trace at $t=0$ as a nonnegative Borel measure $(\mathcal{S} ,u\_{0}),$ where $S$ is the closed set where it is infinite, and $u\_{0}$ is a Radon measure on $Ω\backslash\mathcal{S}.$ We show that the trace is a Radon measure when $q\leqq1.$ For $q\in(1,(N+2)/(N+1)$ and any given Borel measure, we show the existence of a minimal solution, and a maximal one on conditions on $u\_{0}.$ When $\mathcal{S}$ $=\overlineω\capΩ$ and $ω$ is an open subset of $Ω,$ the existence extends to any $q\leqq2$ when $u\_{0}\in L\_{loc}^{1}(Ω)$ and any $q>1$ when $u\_{0}=0$. In particular there exists a self-similar nonradial solution with trace $(\mathbb{R}^{N+},0),$ with a growth rate of order $\left\vert x\right\vert ^{q^{\prime}}$ as $\left\vert x\right\vert \rightarrow\infty$ for fixed $t.$ Moreover we show that the solutions with trace $(\overlineω,0)$ in $Q\_{\mathbb{R}^{N},T}$ may present near $t=0$ a growth rate of order $t^{-1/(q-1)}$ in $ω$ and of order $t^{-(2-q)/(q-1)}$ on $\partial ω.$