Optimal growth rates for a viscous Hamilton-Jacobi equation

The growth of the Lm-norm, m ∈ [1,∞], of non-negative solutions to the Cauchy problem ∂tu  −  Δu  =  |Δu| is studied for non-negative initial data decaying at infinity. More precisely, the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\,\mapsto\,t^{-N/m}||u(t)||m$\end{document} is shown to be bounded from above and from below by positive real numbers. This result indicates an asymptotic behaviour dominated by the hyperbolic Hamilton-Jacobi term of the equation. A one-sided estimate for Δ ln u is also established.