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 language | English (US) |
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Article number | 111639 |
Journal | Nonlinear Analysis, Theory, Methods and Applications |
Volume | 191 |
DOIs | |
State | Published - Feb 2020 |
Bibliographical note
Funding Information:Supported in part by the National Science Foundation Grant DMS-1856491.Supported in part by VEGA Grant 1/0347/18 and by the Slovak Research and Development Agency under the contract No. APVV-14-0378.
Publisher Copyright:
© 2019 Elsevier Ltd
Keywords
- Laguerre polynomials
- Multiplicity of solutions
- Self-similar solutions
- Semilinear heat equation
- Shooting techniques