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

Abstract

Here, we show new apriori 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 $q > 1,$ 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 \[ \left\vert \nabla u(x,t)\right\vert ^{q}\leqq\frac{1}{q-1}\frac{u(x,t)}% {t}\qquad\text{in }Q_{\mathbb{R}^{N},T}. \] Moreover, we prove that the growth of the solutions is limited in space and time: \[ u(x,t)\leqq C(t+t^{-1/(q-1})(1+\left\vert x\right\vert ^{q^{\prime}}% )\qquad\text{in }Q_{\mathbb{R}^{N},T}, \] where $C$ depends on the initial data. We also give existence properties of solutions in $Q_{\Omega,T},$ for initial data locally integrable or unbounded 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.