Non-diffusive large time behaviour for a degenerate viscous Hamilton-Jacobi equation

Abstract

The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem ${\partial }_{t}u={\Delta }_{p}u+|\nabla u{|}^q$ when the initial data converge to zero at infinity. Sufficient conditions on the exponents $p>2$ and $q>1$ are given that guarantee that the diffusion becomes negligible for large times and the $L^{\infty }$-norm of $u(t)$ converges to a positive value as $t\to \infty$.