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Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2015

Global Continuation beyond Singularities on the Boundary for a Degenerate Diffusive Hamilton-Jacobi Equation

Résumé

In this article, we are interested in the Dirichlet problem for parabolic viscous Hamilton-Jacobi Equations. It is well-known that the gradient of the solution may blow up in finite time on the boundary of the domain, preventing a classical extension of the solution past this singularity. This behavior comes from the fact that one cannot prescribe the Dirichlet boundary condition for all time and, in order to define a solution globally in time, one has to use "generalized boundary conditions" in the sense of viscosity solution. In this work, we treat the case when the diffusion operator is the $p$-Laplacian where the gradient dependence in the diffusion creates specific difficulties. In this framework, we obtain the existence and uniqueness of a continuous, global in time, viscosity solution. For this purpose, we prove a Strong Comparison Result between semi-continuous viscosity sub and super-solutions. Moreover, the asymptotic behavior of $\dfrac{u(x; t)}{t}$ is analyzed through the study of the associated ergodic problem.
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Dates et versions

hal-00904365 , version 1 (14-11-2013)

Identifiants

Citer

Amal Attouchi, Guy Barles. Global Continuation beyond Singularities on the Boundary for a Degenerate Diffusive Hamilton-Jacobi Equation. Journal de Mathématiques Pures et Appliquées, 2015, 104 (2), pp.383-402. ⟨hal-00904365⟩
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