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

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 $Q_{Ω,T}=Ω\times\left(0,T\right) ,$ where $q>1,ν\geqq 0,T\in\left(0,\infty\right] ,$ and $Ω=\mathbb{R}^{N}$ or $Ω$ is a smooth bounded domain, and $u_{0}\in L^{r}(Ω),r\geqq1,$ or $u_{0}$ 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 $u_{0}\in \mathcal{M}_{b}(Ω)$ and $q<(N+2)/(N+1),$ or $u_{0}\in L^{r}(Ω)$ 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 $p>1,λ\geqq0,$ and $u$ is a signed solution.

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