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Excluding blowup at zero points of the potential by means of Liouville-type theorems

By Jong-Shenq Guo and Philippe Souplet

Abstract

We consider the diffusive Hamilton-Jacobi equation, with homogeneous Dirichlet conditions and regular initial data. It is known from [Barles-DaLio, 2004] that the problem admits a unique, continuous, global viscosity solution, which extends the classical solution in case gradient blowup occurs. We study the question of the possible loss of boundary conditions after gradient blowup, which seems to have remained an open problem till now. Somewhat surprisingly, our results show that the issue strongly depends on the initial data and reveal a rather rich variety of phenomena. For any smooth bounded domain, we construct initial data such that the loss of boundary conditions occurs everywhere on the boundary, as well as initial data for which no loss of boundary conditions occurs in spite of gradient blowup. Actually, we show that the latter possibility is rather exceptional. More generally, we show that the set of the points where boundary conditions are lost, can be prescribed to be arbitrarily close to any given open subset of the boundary

Topics: [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Publisher: HAL CCSD
Year: 2016
OAI identifier: oai:HAL:hal-01515044v1
Provided by: HAL-Paris 13
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