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

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

2012-02-13

arxiv.org/abs/1202.2674v1

Mathematics

Analysis of PDEs

Scientific paper

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem [{[c]{c}% u_{t}-\nu\Delta u+|\nabla u|^{q}=0u(0)=u_{0}.] in $Q_{\Omega,T}=\Omega\times(0,T) ,$ where $q>1,\nu\geqq 0,T\in (0,\infty ,$ 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 $|x|\rightarrow\infty,$ 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+|u| ^{\lambda-1}u|\nabla u|^{q}=0 \] where $p>1,\lambda\geqq0,$ and $u$ is a signed solution.

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