$L^{\infty}$ estimates and uniqueness results for nonlinear parabolic equations with gradient absorption terms

Abstract

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \[ \left\{\begin{array} [c]{c}% u_{t}-\nu\Delta u+|\nabla u|^{q}=0, u(0)=u_{0}, \end{array} \right. \] in $Q_{\Omega,T}=\Omega\times\left(0,T\right) ,$ where $q>1,\nu\geqq 0,T\in\left(0,\infty\right] ,$ and $\Omega=\mathbb{R}^{N}$ or $\Omega$ is a smooth bounded domain, and $u_{0}\in L^{r}(\Omega),r\geqq1,$ or $u_{0}% \in\mathcal{M}_{b}(\Omega).$ We show $L^{\infty}$ decay estimates, valid for \textit{any weak solution}, \textit{without any conditions a}s $\left\| x\right\| \rightarrow\infty,$ and \textit{without uniqueness assumptions}. As a consequence we obtain new uniqueness results, when $u_{0}\in \mathcal{M}_{b}(\Omega)$ and $q<(N+2)/(N+1),$ or $u_{0}\in L^{r}(\Omega)$ and $q<(N+2r)/(N+r).$ We also extend some decay properties to quasilinear equations of the model type \[ u_{t}-\Delta_{p}u+\left\| u\right\| ^{\lambda-1}u|\nabla u|^{q}=0 \] where $p>1,\lambda\geqq0,$ and $u$ is a signed solution.