Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation
type: Item
instance of: scholarly article
title: Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation
zbMATH Open document ID: 1056.35024
author: Piotr Biler
author: Grzegorz Karch
author: M. Guedda
published in: Journal of Evolution Equations
publication date: 2004-10-28T00:00:00Z
review text: The qualitative properties of nonegative solutions of the Cauchy problem for the viscous Hamilton-Jacobi equation in \((x,t) \in \mathbb R^n \times (0,\infty)\) \[ \begin{aligned} u_t-\Delta u + | \nabla u| ^q & =0, \tag{1}\\ u(x,0) & =u_0(x) \geq 0, \tag{2}\end{aligned} \] under the critical assumption \[ \frac{n+2}{n+1}<q<2 \] are studied. The existence of general selfsimilar solutions of the equation (1), the asymptotic properties of (radially symmetric) selfsimilar solutions of (1) is proved. The behavior of solutions with nonintegrable initial data satisfying, rougly speaking, \(u_0(x) \sim | x| ^{-b}\) as \(| x| \to \infty\) with \(a=\frac{2-q}{q-1}<b<n,\) is established. For the integrable nonegative initial data \(0 \leq u_0 \in L^1(\mathbb R^n)\) the large time asymptotics of solutions is determined by the heat kernel. Namely \(u(t,.) \sim M_{\infty}G(.,t),\) where \[ G(x,t)=(4\pi t)^{-n/2} \exp \biggl(-\frac{| x| ^2}{4t}\biggr), \] and \( M_{\infty}=\lim_{t \to \infty} \int_{\mathbb R^n}u(x,t)dx \) is finite and strictly positive provided \(q>\frac{n+2}{n+1}.\)
reviewed by: Mersaid Aripov
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 2111543
zbMATH Keywords: Cauchy problem
zbMATH Keywords: nonintegrable initial data
zbMATH Keywords: integrable nonegative initial
zbMATH Keywords: large time asymptotics
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2111543