$L^{\infty }$ 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 \[ \left\{\begin{array} [c]{c}% u_{t}-νΔu+|\nabla u|^{q}=0, u(0)=u_{0}, \end{array} \right. \] 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 $u_0\in L^r(\Omega ),r\geqq 1,$ or $u_0$ We show $L^{\infty }$ decay estimates, valid for \textit{any weak solution}, \textit{without any conditions a}s $\Vert x\Vert \to \infty ,$ 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 \[ u_{t}-Δ_{p}u+\left\| u\right\| ^{λ-1}u|\nabla u|^{q}=0 \] where $p>1,\lambda \geqq 0,$ and $u$ is a signed solution.