On the generalized Dirichlet problem for viscous Hamilton--Jacobi equations.
type: Item
instance of: scholarly article
title: On the generalized Dirichlet problem for viscous Hamilton--Jacobi equations.
zbMATH Open document ID: 1056.35071
DOI: 10.1016/S0021-7824(03)00070-9
author: Guy Barles
author: Francesca Da Lio
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2004-05-27T00:00:00Z
review text: The initial and boundary value problem for viscous Hamilton-Jacobi equation \(u_t-\Delta u + | D u| ^p=0, \quad u(x,0)=u_0(x), \quad x \in \Omega, u(x,t)=\phi(x,t),\) where \(\Omega\) is an subset of \(\mathbb R^N\) and \(p>0,\) \(u(x,t)=\phi(x,t)\) on \(\partial \Omega \times (0,\infty)\), \(u_0(x)\) and \(\phi\) are continuons functions satistying the compatiblity conditions, is studied. The authors revisit the results of \textit{M. Fila} and \textit{G. M. Lieberman} [Differ. Integral Equ., 7, No. 3--4, 811--821 (1994; Zbl 0811.35059)], \textit{P. Suplet} [SIAM J. Control Optim. 24, 335-353 (1986)]. It is shown under suitable and natural assumptions on the initial and boundary data the solution of the Dirichlet problem exists for all time, provided that the Dirichlet boundary that condition is understood in the generalized sense of viscosity solutions. This generalized notion of boundary conditions allows loss of boundary conditions, and therefore the authors show that, for \(p > 2,\) such loss of boundary data can actually happen, while the Dirichlet problem can be solved in a classical way for \(1 < p \leq 2\) and even for \(p > 0.\) The authors indicate why a blow up of the gradient or at least some related property is necessary to have such loss of boundary data, explaining the phenomena observed in the above mentioned works.
reviewed by: Mersaid Aripov
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 2066879
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: viscosity solution
zbMATH Keywords: blow up
zbMATH Keywords: blow up of the gradient
MaRDI profile type: Publication
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full work available at URL: s0021-7824(03)00070-9
OpenAlex ID: W2069854767
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description: scientific article; zbMATH DE number 2066879