Global structure of self-similar solutions in a semilinear parabolic equation
type: Item
instance of: scholarly article
title: Global structure of self-similar solutions in a semilinear parabolic equation
zbMATH Open document ID: 0964.35023
author: Munemitsu Hirose
author: Eiji Yanagida
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-12-14T00:00:00Z
review text: This paper is concerned with the structure of solutions of the differential equation \[ u''+ \Biggl({n- 1\over r}+{r\over 2}\Biggr) u'+ {1\over p-1} u+|p|^{p- 1}u= 0,\quad r\in (0,\infty),\quad p>1\tag{1} \] which are related to the radial self-similar solutions of \[ v_t= \Delta v+|v|^{p- 1}v= 0,\quad (t,x)\in (0,\infty)\times \mathbb{R}^n \] via the transformation \[ v(x, t)= t^{-1/(p- 1)}u(r),\quad r= t^{-1/2}|x|. \] With the initial conditions \(u(0)= \alpha\geq 0\), \(u'(0)= 0\) it is known that (1) has a unique global \(C^2\) solution \(u(r; \alpha,p)\). Moreover, the limit \(\lim_{r\to\infty} r^{2/(p- 1)}u(r; \alpha,p)\) exists and is finite. If this limit is zero the solution \(u\) is said to decay rapidly, otherwise it decays slowly. The main result of the paper is the following. For \(k= 0,1,2,\dots\) let \(p_k= 1+2/(n+ 2k)\) and let \(p_{-1}= \infty\) for \(1\leq n\leq 2\), \(n/(n-2)\) for \(n> 2\). Then there are positive \(C^1\) functions \(\alpha_k\) defined for \(p\in (p_k, p_{-1}]\) such that \(u(r; \alpha,p)\) is rapidly decaying for \(\alpha= \alpha_k(p)\) and has precisely \(k\) zeros in \((0,\infty)\). Moreover, for fixed \(p\in (p_k, p_{k-1}]\) the sequence \(\alpha_k(p)\) is increasing and unbounded with \(k\). Finally, \(u(r;\alpha, p)\) is slowly decaying with \(k\) positive zeros for \(0< \alpha< \alpha_k(p)\), slowly decaying with \(i\) positive zeros for \(\alpha_i(p)< \alpha< \alpha_{i+1}(p)\), \(i= k,k+1,k+2\dots\;\). A similar result is proved for the Dirichlet initial conditions \(u(0)= 0\), \(u'(0)= \beta\geq 0\) for the one-dimensional problem \(n= 1\); now \(p_k= 1+ 1/(1+k)\). The proofs rely mainly on known results and counting arguments using the Sturm comparison theorem. The final section deals with solutions of (1) on the whole real line. It is shown that for each \(p> p_k= 1+ 2/(k+1)\) there are exactly two solutions of (1) which have precisely \(k\) zeros and decay rapidly at \(\infty\).
reviewed by: J. Graham-Eagle
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1449212
zbMATH Keywords: Sturm comparison theorem
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6706
OpenAlex ID: W2033492569
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DOI: 10.1006/JMAA.2000.6706
description: scientific article; zbMATH DE number 1449212