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

Abstract : 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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Journal de Mathématiques Pures et Appliquées, Elsevier, 2015, 104 (2), pp.383-402. 〈http://www.journals.elsevier.com/journal-de-mathematiques-pures-et-appliquees〉
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  • HAL Id : hal-00904365, version 1
  • ARXIV : 1311.3593

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Amal Attouchi, Guy Barles. Global Continuation beyond Singularities on the Boundary for a Degenerate Diffusive Hamilton-Jacobi Equation. Journal de Mathématiques Pures et Appliquées, Elsevier, 2015, 104 (2), pp.383-402. 〈http://www.journals.elsevier.com/journal-de-mathematiques-pures-et-appliquees〉. 〈hal-00904365〉

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