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Pré-Publication, Document De Travail Année : 2015

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

Résumé

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 $\partial \omega.$
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Dates et versions

hal-01024881 , version 1 (16-07-2014)
hal-01024881 , version 2 (20-08-2014)
hal-01024881 , version 3 (12-02-2015)

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Marie-Françoise Bidaut-Véron, Nguyen Anh Dao. Initial trace of solutions of Hamilton-Jacobi parabolic equation with absorption. 2015. ⟨hal-01024881v3⟩
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