Local and global estimates of solutions of Hamilton-Jacobi parabolic equation with absorption

Abstract

We obtain new a priori estimates for the nonnegative solutions of the equation \[ u_{t}-Δu+|\nabla u|^{q}=0 \] 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 $\partial \Omega \times \left(0,T\right).$ In case $\Omega ={\mathbb{R}}^N,$ we show that any solution $u\in C^{2,1}(Q_{{\mathbb{R}}^N,T})$ of equation (1.1) in $Q_{{\mathbb{R}}^N,T}$ (in particular any weak solution if $q\leqq 2),$ without condition as $\left|x\right|\to \infty ,$ satisfies the universal estimate \[ \left\vert \nabla u(.,t)\right\vert ^{q}\leqq\frac{1}{q-1}\frac{u(.,t)}% {t},\qquad\text{in }Q_{\mathbb{R}^{N},T}. \] Moreover we prove that the growth of $u$ is limited by $C(t+t^{-1/(q-1}$ where $C$ depends on $u.$ We also give existence properties of solutions in $Q_{\Omega ,T},$ for initial data locally integrable or even unbounded Radon measures. We give a nonuniqueness result in case $q>2.$ Finally we show that besides the local regularizing effect of the heat equation, $u$ satisfies a second effect of type $L_{loc}^R$ due to the gradient term.