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

P. Poláčik, P. Quittner

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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
StatePublished - Feb 2020



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

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