Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation

TL;DR

This paper proves that radially symmetric solutions to a degenerate diffusive Hamilton-Jacobi equation with specific boundary conditions converge to a steady state, characterizing these solutions and analyzing their long-term behavior.

Contribution

It establishes convergence to steady states for radially symmetric solutions and characterizes stationary solutions for a class of degenerate diffusive Hamilton-Jacobi equations.

Findings

Radially symmetric solutions converge to a steady state.

Stationary solutions are explicitly characterized.

Results apply to p-Laplacian with p ≥ 2.

Abstract

Convergence to a single steady state is shown for non-negative and radially symmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions, the diffusion being the $p$-Laplacian operator, $p\ge 2$, and the source term a power of the norm of the gradient of $u$. As a first step, the radially symmetric and non-increasing stationary solutions are characterized.