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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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Submitted on : Thursday, November 14, 2013 - 1:10:30 PM
Last modification on : Tuesday, January 11, 2022 - 5:56:07 PM
Long-term archiving on: : Saturday, February 15, 2014 - 4:33:15 AM

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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. ⟨hal-00904365⟩

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