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

Here we study the initial trace problem for the nonnegative solutions of the equation $$u\mathrm{\_}t-\mathrm{\Delta }u+\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q=0$$ in $Q\mathrm{\_}\mathrm{\Omega },T=\mathrm{\Omega }\times (0,T),$ $T\leqq \mathrm{\infty },$ where $q>0,$ and $\mathrm{\Omega }={\mathbb{R}}^N,$ or $\mathrm{\Omega }$ is a smooth bounded domain of ${\mathbb{R}}^N$ and $u=0$ on $\mathrm{\partial }\mathrm{\Omega }\times (0,T)\mathrm{.}$ We can define the trace at $t=0$ as a nonnegative Borel measure $($\mathcal{S}$ ,u_{0}),$where$S$istheclosedsetwhereitisinfinite,and$u_{0}$ is a Radon measure on $\mathrm{\Omega }\mathrm{\backslash }\mathcal{S}\mathrm{.}$ We show that the trace is a Radon measure when $q\leqq 1.$ For $q\in (1,(N+2)\mathrm{/}(N+1)$ and any given Borel measure, we show the existence of a minimal solution, and a maximal one on conditions on $u\mathrm{\_}0.$ When $\mathcal{S}$ $=\bar{\omega }\cap \mathrm{\Omega }$ and $\omega$ is an open subset of $\mathrm{\Omega },$ the existence extends to any $q\leqq 2$ when $u\mathrm{\_}0\in L\mathrm{\_}{loc}^1(\mathrm{\Omega })$ and any $q>1$ when $u\mathrm{\_}0=0$. 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$\left\vert x\right\vert \rightarrow\infty$forfixed$t.$Moreoverweshowthatthesolutionswithtrace$(\overline{\omega},0)$in$Q_{$\mathbb{R}$^{N},T}$maypresentnear$t=0$agrowthrateoforder$t^{-1/(q-1)}$in$\omega$andoforder$t^{-(2-q)/(q-1)}$on$\partial \omega.$