# 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.

## Key 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.