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

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Contribution

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Abstract

We obtain new a priori estimates 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 $\partial\Omega\times\left( 0,T\right) .$ 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 $q\leqq2),$ without condition as $\left\vert x\right\vert \rightarrow\infty,$ 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 $C(t+t^{-1/(q-1}% )(1+\left\vert x\right\vert ^{q^{\prime}}),$ 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 $L_{loc}^{R}% -L_{loc}^{\infty},$ due to the gradient term.