Self-similar extinction for a diffusive Hamilton-Jacobi equation with critical absorption

TL;DR

This paper characterizes the extinction behavior of solutions to a critical diffusive Hamilton-Jacobi equation, showing convergence to a unique self-similar profile with a variational structure and classifying all such solutions.

Contribution

It introduces a detailed analysis of the extinction profile for the equation, establishing convergence to a unique self-similar solution and classifying all possible profiles.

Findings

Solutions approach a self-similar form near extinction time.

The profile solving the associated ODE is unique and decays exponentially.

A variational structure and Pohozaev functional are key to the proofs.

Abstract

The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption $\partial_t u - \Delta_p u + |\nabla u|^{p-1} = 0$ in $(0, \infty) \times \mathbb{R}^N$, and fast diffusion $2N/(N + 1) < p < 2$. Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as $|x| \to \infty$, it is shown that the corresponding solution $u$ to the above equation approaches a uniquely determined separate variable solution of the form $U(t, x) = (T_e - t)^{1/(2-p)} f_* (|x|), (t, x) \in (0, T_e) \times \mathbb{R}^N$, as $t \to T_e$, where $T_e$ denotes the finite extinction time of $u$. A cornerstone of the convergence proof is an underlying variational structure of the equation. Also, the selected profile $f_*$ is the unique non-negative solution to a second order ordinary differential equation which decays exponentially at infinity. A complete classification of solutions to this equation is provided, thereby describing all separate variable solutions of the original equation. One important difficulty in the uniqueness proof is that no monotonicity argument seems to be available and it is overcome by the construction of an appropriate Pohozaev functional.