Large-time behavior of unbounded solutions of the viscous Hamilton-Jacobi equation: quadratic and subquadratic cases

TL;DR

This paper analyzes the long-term behavior of solutions to the viscous Hamilton-Jacobi equation with unbounded initial data in quadratic and subquadratic cases, overcoming challenges due to the lack of comparison principles.

Contribution

It establishes the large-time behavior for any solution of the viscous Hamilton-Jacobi equation in the specified cases, despite the absence of solution uniqueness.

Findings

Large-time behavior characterized for unbounded solutions

Generalized simplicity holds for ergodic problem solutions

Results apply even with multiple solutions present

Abstract

We determine the large-time behavior of unbounded solutions for the so-called viscous Hamilton Jacobi equation, $u_t - \Delta u + |Du|^m = f(x)$, in the quadratic and subquadratic cases (i.e., for $1<m\leq 2$), with a particular focus on allowing arbitrary growth at infinity for $f$ and the prescribed initial data. The lack of a comparison principle for the associated ergodic problem is overcome by proving that a generalized simplicity holds for sub- and supersolutions of the ergodic problem. Moreover, as the uniqueness of solutions of the parabolic problem remains open in the current setting, our result on large-time holds for any solution, even if multiple solutions exist.