Here we study the initial trace problem for the nonnegative solutions of the equation \[ u\_{ t}-\Delta u+|\nabla u|^{ q}=0 \] in $Q\_{ \Omega,T}=\Omega\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $\Omega=\mathbb{ R}^{ N},$ or $\Omega$ is a smooth bounded domain of $\mathbb{ R}^{ N}$ and $u=0$ on $\partial\Omega\times\left( 0,T\right) .$ 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 $\Omega\backslash\mathcal{ S}.$ 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 $\mathcal{ S}$ $=\overline{ \omega}\cap\Omega$ and $\omega$ is an open subset of $\Omega,$ the existence extends to any $q\leqq2$ when $u\_{ 0}\in L\_{ loc}^{ 1}(\Omega)$ 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 $(\overline{ \omega},0)$ in $Q\_{ \mathbb{ R}^{ N},T}$ may present near $t=0$ a growth rate of order $t^{ -1/(q-1)}$ in $\omega$ and of order $t^{ -(2-q)/(q-1)}$ on $\partial \omega.$
Categories: math.AP