Mathematics – Analysis of PDEs
Scientific paper
2009-12-17
Journal de Math\'ematiques Pures et Appliqu\'ees (9) 94, 5 (2010) 497-519
Mathematics
Analysis of PDEs
Scientific paper
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.
Barles Guy
Porretta Alessio
Tabet Tchamba Thierry Wilfried
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