Decay estimates for a viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions
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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.
Benachour, S., Dăbuleanu-Hapca, S. and Laurençot, P. 2007. Decay estimates for a viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions. Asymptotic Analysis. 51(3-4): pp.209-229. [Online] Available from: http://content.iospress.com/articles/asymptotic-analysis/asy802