Self-similar solutions for a class of non-divergence form equations
type: Item
instance of: scholarly article
title: Self-similar solutions for a class of non-divergence form equations
zbMATH Open document ID: 1319.35084
author: Chunhua Jin
author: Jingxue Yin
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2013-07-19T00:00:00Z
review text: This paper deals with self-similar solutions for degenerate and singular parabolic equations in non divergence form of the type \[ \frac{\partial u}{\partial t}= u^{m}\mathrm{div}(|\nabla u|^{p-2}\nabla u),\tag{1} \] where \( m\geq 1 \) and \( p>1 \). It may be verified that if \( u(x,t)\not\equiv 0 \) is a solution of (1), then \[ u_{\lambda}(x,t)= \lambda^{\alpha}u( \lambda^{-\beta}x,\lambda t),\quad \beta\geq 0, \alpha=\frac{1+\beta p}{m+p-2}, \] for \( \lambda > 0\), are solutions of (1). In particular, if \( \lambda=1/t\), then \( u_{\lambda}(x,t)= t^{-\alpha}f( t^{\beta}x )\) is called a self-similar solution of (1). Generally, the study of self-similar solutions is restricted to the forward self-similar solution \[ u(x,t)=(t+\tau)^{-\alpha}f(\eta), \] where \( \eta=(t+\tau)^{\beta}x \) and to the backward self-similar solution \[ u(x,t)=(\tau -t)^{-\alpha}f(\eta),\eta=(\tau -t)^{\beta}x. \] In this paper, a family of forward self-similar solutions of (1), radially symmetric in \(x\), of the form \( u(x,t)=(t+1)^{-\alpha}f(\eta) \) are obtained, where \( \eta =(t+1)^{\beta}|x|^{2}, \beta\geq 0, \alpha=\frac{2+p\beta}{2(m+p-2)}. \) It is verified that \( u(x,t) \) is a self-similar solution of (1) if and only if \( f \) satisfies the equation \[ 2^{p}\eta^{p/2}f^{m}(|f'|^{p-2}f')'+(N+p-2)2^{p-1}\eta^{(p-2)/2}f^{m}|f'|^{p-2}f'-\beta\eta f'+\alpha f=0. \] For \( h=\eta^{p/2}|f'|^{p-2}f', \) (2) is transformed into \[ f'=\eta^{-\frac{p}{2(p-1)}}|h|^{q-2}h, h'=\frac{2-N}{2\eta}h + 2^{-p}\beta \eta^{(p-2)/2(p-1)}f^{-m}|h|^{q-2}h-\frac{\alpha}{2^{p}}f^{1-m}, \] where \( q \) is the dual number of the constant \( p \), that is, \( 1/p + 1/q = 1. \) Thus, the system (3) is considered with the initial value conditions \[ f(0)=A, \quad A>0, h(0)=0. \] In this paper, the solutions with compact support of (3) and (4) are obtained. The authors discuss convergence speeds of the solutions going to zero. Furthermore, they investigate the singularities of the reciprocal of solu…
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35C06
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 6189421
zbMATH Keywords: self-similar solution
zbMATH Keywords: non divergence form
zbMATH Keywords: convergence rates
MaRDI profile type: Publication
full work available at URL: s00030-012-0185-y
OpenAlex ID: W2048040880
cites work: Persistence and extinction in single-species reaction-diffusion models
cites work: Blow-up of solutions of nonlinear degenerate parabolic equations
cites work: Blow-up for solutions of some degenerate parabolic equations
cites work: Q4343800
cites work: On the formation of singularities in the curve shortening flow
cites work: A degenerate diffusion problem not in divergence form
cites work: On a class of similarity solutions of the porous media equation
cites work: On a class of similarity solutions of the porous media equation. II
cites work: On a class of similarity solutions of the porous media equation. III
cites work: Variational problems related to self-similar solutions of the heat equation
cites work: Radial symmetry of self-similar solutions for semilinear heat equations
cites work: Shrinking self-similar solutions of a nonlinear diffusion equation with nondivergence form.
cites work: Asymptotically self‐similar blow‐up of semilinear heat equations
cites work: Periodic solutions of a non-divergent diffusion equation with nonlinear sources
DOI: 10.1007/S00030-012-0185-Y
description: scientific article; zbMATH DE number 6189421