Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations

By S. Benachour, Grzegorz Karol Karch, Philippe Laurencçot. 2004

60 citations recorded by the source corpus.

The large time behavior of solutions to the Cauchy problem for the viscous Hamilton–Jacobi equation ut−Δu+|∇u|q=0 is classified. If q>qc:=(N+2)/(N+1), it is shown that non-negative solutions corresponding to integrable initial data converge in W1,p(RN) as t→∞ toward a multiple of the fundamental solution for the heat equation for every p∈[1,∞] (diffusion-dominated case). On the other hand, if 1<q<qc, the large time asymptotics is given by the very singular self-similar solution of the viscous Hamilton–Jacobi equation. For non-positive and integrable solutions, the large time behavior of solutions is more complex. The case q⩾2 corresponds to the diffusion-dominated case. The diffusion profiles in the large time asymptotics appear also for qc<q<2 provided suitable smallness assumptions are imposed on the initial data. Here, however, the most important result asserts that under some conditions on initial data and for 1<q<2, the large time behavior of solutions is given by the self-similar viscosity solutions to the non-viscous Hamilton–Jacobi equation zt+|∇z|q=0 supplemented with the initial datum z(x,0)=0 if x≠0 and z(0,0)<0.

Subfields: Computational Theory and Mathematics, Finance, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/bd78afb57b7bc9b3a4718369b175e113891b83d0

Citations in the corpus

• Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains (2010, unknown): “Asymptotic behavior of global solutions of (1.5)‐(1.7) and of the corresponding Cauchy problem is investigated in [2,4,9, 10 ,21,30,33,41,42] and references therein.”