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 \(t\to\infty\).