Complete classification of shape functions of self-similar solutions
type: Item
instance of: scholarly article
title: Complete classification of shape functions of self-similar solutions
zbMATH Open document ID: 1129.35016
author: Zhongbo Fang
author: Minkyu Kwak
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-04T00:00:00Z
review text: Let \(q>0\), \(q\neq1\), and consider a generalized KPZ equation of the type \(u_t=u_{xx}-| u_x| ^q\), \(x>0\), \(t>0\), complemented by the Neumann boundary condition \(u_x(0,t)=0\). Self-similar solutions of this problem are of the form \(u(x,t)=t^{-\alpha}g(\xi)\), where \(\alpha=(2-q)/(2q-2)\), \(\xi=x/\sqrt{t}\) and \(g\) solves the ODE \(g''+{\xi\over2}g'+\alpha g=| g'| ^q\). The authors study the asymptotic behavior of solutions of this ODE subject to the initial conditions \(g'(0)=0\), \(g(0)=\lambda>0\). In particular, they are interested in the existence of fast orbits (satisfying \(\lim_{\xi\to\infty}\xi^{2\alpha}g(\xi)=0\)) which correspond to very singular solutions \(u\) of the PDE. The results depend on the exponent \(q\); the critical values of \(q\) are \(1,3/2\) and \(2\).
reviewed by: Pavol Quittner
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 5149537
zbMATH Keywords: very singular solution
zbMATH Keywords: blow-up
zbMATH Keywords: one space dimension
zbMATH Keywords: generalized KPZ equatio
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: fast orbits
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.08.042
OpenAlex ID: W2069903433
cites work: Asymptotic estimates of solutions of \(u_ t - \frac12\Delta u =-|\nabla u|\) in \(\mathbb{R}_ + \times \mathbb{R}^ d,\quad d\geq 2\)
cites work: Very singular solutions to a nonlinear parabolic equation with absorption. I: Existence
cites work: Very singular solutions to a nonlinear parabolic equation with absorption II. Uniqueness
cites work: Asymptotic profiles of solutions to viscous Hamilton--Jacobi equations
cites work: Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation
cites work: A Neumann problem for a convection-diffusion equation on the half-line
cites work: A very singular solution of the heat equation with absorption
cites work: Large time behavior of solutions of a dissipative semilnear heat equation
cites work: The Cauchy problem for \(u_{t}=\Delta u+| \nabla u|^q\)
cites work: Self-similar solutions of convection-diffusion processes
cites work: Self-similar solutions to the generalized deterministic KPZ equation
cites work: A semilinear heat equation with singular initial data
cites work: A porous media equation with absorption. II: Uniqueness of the very singular solution
cites work: A porous media equation with absorption. I: Long time behaviour
cites work: Dynamic Scaling of Growing Interfaces
cites work: A very singular solution and other self-similar solutions of the heat equation with convection
cites work: A very singular solution of the porous media equation with absorption
DOI: 10.1016/J.JMAA.2006.08.042
description: scientific article; zbMATH DE number 5149537