Finite extinction time for some perturbed Hamilton-Jacobi equations
type: Item
instance of: scholarly article
title: Finite extinction time for some perturbed Hamilton-Jacobi equations
zbMATH Open document ID: 0767.49020
DOI: 10.1007/BF01182596
author: S. Singh
published in: Applied Mathematics and Optimization
publication date: 1993-04-01T00:00:00Z
review text: A class of Hamilton-Jacobi like equations \(u_ t-R|\nabla u|^ m+\lambda u^ q=0\) are considered with initial condition \(u_ 0(\cdot)=u(\cdot,0)\) satisfying \(-R|\nabla u_ 0|^ m+\lambda u_ 0^ q\geq 0\). It is shown that these equations have minimum non- negative viscosity solutions which are zero after some time \(t_ \infty(x)\), i.e. \(u(x,t)>0\) if \(0<t<t_ \infty(x)\), \(u(x,t)=0\) if \(t\geq t_ \infty(x)\). Differentiability of \(u\) at the support boundary is shown and the Lipschitz continuity of \(t_ \infty(x)\) is also proved. The extinction rate of \(u\) is considered in detail and under certain conditions it is shown that the exact value can be obtained. Finally the representation formula \(u(x,t)=\max\{\gamma^{-1}(\gamma(u_ 0(x-\xi t))-\lambda t):\;|\xi|\leq R\}\) is obtained.
Mathematics Subject Classification ID: 49L99
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35C99
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 147398
zbMATH Keywords: perturbed Hamilton-Jacobi equations
zbMATH Keywords: minimum non-negative viscosity solutions
zbMATH Keywords: extinction rate
reviewed by: S. P. Banks
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 147398