Decay estimates for a viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions

Saïd Benachour, Simona M. Hapca, Philippe Laurençot

Research output: Contribution to journalArticle

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Abstract

Global classical solutions to the viscous Hamilton–Jacobi equation ut−Δu=a|∇u|p in (0,∞)×Ω with homogeneous Dirichlet boundary conditions are shown to converge to zero in W1,∞(Ω) at the same speed as the linear heat semigroup when p>1. For p=1, an exponential decay to zero is also obtained but the rate depends on a and differs from that of the linear heat equation. Finally, if p∈(0,1) and a<0, finite time extinction occurs for non-negative solutions.
Original languageEnglish
Pages (from-to)209-229
Number of pages21
JournalAsymptotic Analysis
Volume51
Issue number3-4
StatePublished - 2007

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Hamilton-Jacobi equation
Dirichlet boundary conditions
Zero
Extinction time
Heat semigroup
Global classical solution
Decay estimates
Nonnegative solution
Exponential decay
Heat equation
Linear equation
Converge

Cite this

Benachour, Saïd; Hapca, Simona M.; Laurençot, Philippe / Decay estimates for a viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions.

In: Asymptotic Analysis, Vol. 51, No. 3-4, 2007, p. 209-229.

Research output: Contribution to journalArticle

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Decay estimates for a viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions. / Benachour, Saïd; Hapca, Simona M.; Laurençot, Philippe.

In: Asymptotic Analysis, Vol. 51, No. 3-4, 2007, p. 209-229.

Research output: Contribution to journalArticle

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AU - Benachour,Saïd

AU - Hapca,Simona M.

AU - Laurençot,Philippe

PY - 2007

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N2 - Global classical solutions to the viscous Hamilton–Jacobi equation ut−Δu=a|∇u|p in (0,∞)×Ω with homogeneous Dirichlet boundary conditions are shown to converge to zero in W1,∞(Ω) at the same speed as the linear heat semigroup when p>1. For p=1, an exponential decay to zero is also obtained but the rate depends on a and differs from that of the linear heat equation. Finally, if p∈(0,1) and a<0, finite time extinction occurs for non-negative solutions.

AB - Global classical solutions to the viscous Hamilton–Jacobi equation ut−Δu=a|∇u|p in (0,∞)×Ω with homogeneous Dirichlet boundary conditions are shown to converge to zero in W1,∞(Ω) at the same speed as the linear heat semigroup when p>1. For p=1, an exponential decay to zero is also obtained but the rate depends on a and differs from that of the linear heat equation. Finally, if p∈(0,1) and a<0, finite time extinction occurs for non-negative solutions.

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JO - Asymptotic Analysis

T2 - Asymptotic Analysis

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