### 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 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 191, 111639. https://doi.org/10.1016/j.na.2019.111639

}

TY - JOUR

T1 - On the multiplicity of self-similar solutions of the semilinear heat equation

AU - Poláčik, P.

AU - Quittner, P.

PY - 2020/2

Y1 - 2020/2

N2 - 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.

AB - 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.

KW - Laguerre polynomials

KW - Multiplicity of solutions

KW - Self-similar solutions

KW - Semilinear heat equation

KW - Shooting techniques

UR - http://www.scopus.com/inward/record.url?scp=85072560825&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072560825&partnerID=8YFLogxK

U2 - 10.1016/j.na.2019.111639

DO - 10.1016/j.na.2019.111639

M3 - Article

AN - SCOPUS:85072560825

VL - 191

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

M1 - 111639

ER -