| Contents |
| Here we consider the nonnegative solutions of the parabolic Hamilton-Jacobi
equation
\[
u_{t}-\Delta u+\left\vert \nabla u\right\vert ^{q}=0
\]
in $\Omega\times\left( 0,T\right) $ where $\Omega=\mathbb{R}^{N}$ or
$\Omega$ is a bounded domain of $\mathbb{R}^{N},$ $q>0.$
We give new a priori local or global estimates for solutions, without
conditions as $\left\vert x\right\vert \rightarrow\infty$ or $x\rightarrow
\partial\Omega,$ and corresponding existence results with initial data measure.
We study the existence of an initial trace. We show that all the solutions
admit a trace as a Borel measure $(S,u_{0}):$ there exist a set $\mathcal{S}%
\subset\Omega$ such that $\mathcal{R}=\Omega\backslash\mathcal{S}$ is open,
and a (possibly unbounded) measure $u_{0}\in\mathcal{M}^{+}(\mathcal{R})$,
such that
\[
\lim_{t\rightarrow0}\int_{\mathcal{R}}u(.,t)\psi=\int_{\mathcal{R}}\psi
d\mu_{0},\qquad\forall\psi\in C_{c}^{0}(\mathcal{R}),
\]
\medskip\
\[
\lim_{t\rightarrow0}\int_{\mathcal{U\cap S}}u(.,t)dx=\infty,\qquad
\forall\mathcal{U}\text{ open}\subset\Omega,\text{s.th. }\mathcal{U\cap
S\neq\emptyset}.
\]
We give more generally existence results of solutions with such a trace,
according to assumptions on $u_{0},$ and give their behaviour as $\left\vert
x\right\vert \rightarrow\infty.$ In particular we construct a solution with
trace $(\mathbb{R}^{N+},0).$ When $q\le1,$ we show that $\mathcal{S}$ is empty. |
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