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Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2010

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

Résumé

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.
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Dates et versions

hal-00441975 , version 1 (17-12-2009)

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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, 2010, (9) 94 (5), pp.497-519. ⟨hal-00441975⟩
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