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

TL;DR

This paper studies the long-term behavior of solutions to a degenerate viscous Hamilton-Jacobi equation, showing conditions under which solutions become non-diffusive and stabilize to a positive value over time.

Contribution

It provides new sufficient conditions on exponents p and q that ensure the diffusion term becomes negligible and solutions converge to a non-zero limit.

Findings

Diffusion becomes negligible for large times under certain conditions.

Solutions' L-infinity norm converges to a positive value.

Identifies specific exponent ranges for non-diffusive behavior.

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 $t\to\infty$.