On the multiplicity of self-similar solutions of the semilinear heat equation

P. Poláčik, P. Quittner

Research output: Contribution to journalArticle

Abstract

In studies of superlinear parabolic equations ut=Δu+up,x∈RN,t>0,where p>1, backward self-similar solutions play an important role. These are solutions of the form u(x,t)=(T−t)−1∕(p−1)w(y), where y≔x∕T−t, T is a constant, and w is a solution of the equation Δw−y⋅∇w∕2−w∕(p−1)+wp=0. We consider (classical) positive radial solutions w of this equation. Denoting by pS, pJL, pL the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for p∈(pS,pJL) there are only countably many solutions, and for p∈(pJL,pL) there are only finitely many solutions. This result answers two basic open questions regarding the multiplicity of the solutions.

Original languageEnglish (US)
Article number111639
JournalNonlinear Analysis, Theory, Methods and Applications
Volume191
DOIs
StatePublished - Feb 2020

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Semilinear Heat Equation
Self-similar Solutions
Multiplicity
Positive Radial Solutions
Parabolic Equation
Exponent
Hot Temperature
Form

Keywords

  • Laguerre polynomials
  • Multiplicity of solutions
  • Self-similar solutions
  • Semilinear heat equation
  • Shooting techniques

Cite this

On the multiplicity of self-similar solutions of the semilinear heat equation. / Poláčik, P.; Quittner, P.

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 191, 111639, 02.2020.

Research output: Contribution to journalArticle

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