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

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.
Keywords :
Document type :
Journal articles

Cited literature [16 references]

https://hal.archives-ouvertes.fr/hal-00441975
Contributor : Guy Barles <>
Submitted on : Thursday, December 17, 2009 - 5:18:09 PM
Last modification on : Saturday, April 13, 2019 - 1:22:35 AM
Document(s) archivé(s) le : Thursday, June 17, 2010 - 10:00:19 PM

### Files

Files produced by the author(s)

### Identifiers

• HAL Id : hal-00441975, version 1
• ARXIV : 0912.3506

### Citation

Guy Barles, Alessio Porretta, Thierry Wilfried Tabet Tchamba. On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations. Journal de Mathématiques Pures et Appliquées, Elsevier, 2010, (9) 94 (5), pp.497-519. ⟨hal-00441975⟩

Record views