Asymptotic profiles of solutions to viscous Hamilton–Jacobi equations

Abstract

The large time behavior of solutions to the Cauchy problem for the viscous Hamilton-Jacobi equation u t − ∆u + |∇u| q = 0 is classified. If q > q c := (N + 2)/(N + 1), it is shown that non-negative solutions corresponding to integrable initial data converge in W 1,p (R N ) 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 < q c , the large time asymptotics is given by the very singular self-similar solutions of the viscous Hamilton-Jacobi equation.2000 Mathematics Subject Classification: 35K15, 35B40. RésuméNous classifions le comportement asymptotique des solutions du problème de Cauchy pour l'équation de Hamilton-Jacobi avec diffusion u t −∆u+|∇u| q = 0. Si q > q c := (N +2)/(N +1), nous montrons que, lorsque t → ∞, les solutions intégrables et positives convergent dans W 1,p (R N ) vers un multiple de la solution fondamentale de l'équation de la chaleur pour tout p ∈ [1, ∞] (diffusion dominante). Ensuite, si 1 < q < q c , le comportement asymptotique est décrit par la solution très singulière auto-similaire de l'équation de Hamilton-Jacobi avec diffusion.En ce qui concerne les solutions intégrables et négatives, la situation est plus complexe. Le terme de diffusion est de nouveau dominant si q ≥ 2, ainsi que lorsque q c < q < 2 pourvu que la donnée initiale soit suffisamment petite. Ensuite, pour 1 < q < 2, nous identifions une classe de données initiales pour laquelle le comportement asymptotique des solutions est donné par une solution de viscosité auto-similaire de l'équation de Hamilton-Jacobi z t + |∇z| q = 0 avec la condition initiale (non continue) z(x, 0) = 0 si x = 0 et z(0, 0) < 0.

Localized Non-diffusive Asymptotic Patterns for Nonlinear Parabolic Equations with Gradient Absorption

“…[11]. The pattern that we see after the corresponding renormalization, F q (y), is a modification of the Gaussian profile corresponding to the Very Singular Solution and changes with q.…”

“…Assertions (10), (11), and (12) (13) for u(t) ∞ and ∇u(t) ∞ follow then from that for u(t) 1 by (11) with s = t/2. …”

“…To be more precise, in the upper range q > q * , the large time dynamics is governed by the sole diffusion term and u behaves as a multiple of the fundamental solution to the linear heat equation [11,17]. After convenient renormalization we thus see the typical bell-shaped Gaussian profile, in the sense that t N/2 u(t, x) → F (y) = c e −y 2 /2 with y = x/t 1/2 , just as the same form as in the case with no absorption.…”

Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent

“…The equation (1.1) has been studied by various authors for different values of the parameters p 2 and q > 1 as a model of linear or nonlinear diffusion with gradient-dependent absorption: see [8,9,11,12,15] for the semilinear case p = 2, and [1,7,16,20] for the quasilinear case p > 2. It has been shown that the large-time behaviour of this initial-value problem depends on the relative influence of the diffusion and absorption terms and leads to a classification into the following ranges of q:…”

Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent

“…As a matter of fact, if q > q ⋆ it is shown in [3,13] that the nonlinear term |∇u| q becomes negligible for large times, and that the solution of (1) behaves as t → ∞ like the self-similar solution I ∞ g of the linear heat equation, where g(t, x) = 1 t N/2 G x t 1/2 and G(ξ) = 1 (4π) N/2 exp − |ξ| 2 …”