### Abstract

In studies of superlinear parabolic equations u_{t}=Δu+u^{p},x∈R^{N},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)+w^{p}=0. We consider (classical) positive radial solutions w of this equation. Denoting by p_{S}, p_{JL}, p_{L} the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for p∈(p_{S},p_{JL}) there are only countably many solutions, and for p∈(p_{JL},p_{L}) there are only finitely many solutions. This result answers two basic open questions regarding the multiplicity of the solutions.

Original language | English (US) |
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Article number | 111639 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 191 |

DOIs | |

State | Published - Feb 2020 |

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### Keywords

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