On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations

TL;DR

This paper investigates the long-term behavior of solutions to subquadratic viscous Hamilton-Jacobi equations with Dirichlet boundary conditions, revealing different asymptotic behaviors depending on the growth rate of the gradient and the ergodic constant.

Contribution

It extends the understanding of large time behavior for subquadratic cases, showing new phenomena when the gradient growth is at or below a critical threshold.

Findings

For m > 3/2 and c > 0, solutions behave similarly to the superquadratic case.

When m ≤ 3/2 or c = 0, solutions can become unbounded from below after adjusting by ct.

Different asymptotic behaviors are characterized based on the gradient growth and ergodic constant.

Abstract

In this article, we are interested in the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi Equations. In the superquadratic case, the third author has proved that these solutions can have only two different behaviors: either the solution of the evolution equation converges to the solution of the associated stationary generalized Dirichlet problem (provided that it exists) or it behaves like $-ct+\varphi (x)$ where $c\geq0$ is a constant, often called the "ergodic constant" and $\varphi$ is a solution of the so-called "ergodic problem". In the present subquadratic case, we show that the situation is slightly more complicated: if the gradient-growth in the equation is like $|Du|^m$ with $m>3/2,$ then analogous results hold as in the superquadratic case, at least if $c>0.$ But, on the contrary, if $m\leq 3/2$ or $c=0,$ then another different behavior appears since $u(x,t) + ct$ can be unbounded from below where $u$ is the solution of the subquadratic viscous Hamilton-Jacobi Equations.