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Existence and multiplicity of self-similar solutions for heat equations with nonlinear boundary conditions

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Abstract

This paper is concerned with self-similar solutions in the half-space for linear and semilinear heat equations with nonlinear boundary conditions. Existence, multiplicity and positivity of these solutions are analyzed. Self-similar profiles are obtained as solutions of a nonlinear elliptic PDE with drift term and a nonlinear Neummann boundary condition. For that, we employ a variational approach and derive some compact weighted embeddings for the trace operator.

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Acknowledgments

The authors are indebted to the anonymous referee for his/her suggestions which improved the presentation of the paper.

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Authors and Affiliations

  1. IMECC-Departamento de Matemática, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Campinas, SP, CEP 13083-859, Brazil

    Lucas C. F. Ferreira

  2. Departamento de Matemática, Universidade de Brasília, Brasília, DF, CEP 70910-900, Brazil

    Marcelo F. Furtado

  3. Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, PB, CEP 58051-900, Brazil

    Everaldo S. Medeiros

Authors
  1. Lucas C. F. Ferreira
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  2. Marcelo F. Furtado
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  3. Everaldo S. Medeiros
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Corresponding author

Correspondence to Lucas C. F. Ferreira.

Additional information

Communicated by P. Rabinowitz.

L. C. F. Ferreira was supported by FAPESP and CNPq, Brazil. M. F. Furtado was supported by CNPq, Brazil. E. S. Medeiros was supported by CNPq, Brazil.

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Ferreira, L.C.F., Furtado, M.F. & Medeiros, E.S. Existence and multiplicity of self-similar solutions for heat equations with nonlinear boundary conditions. Calc. Var. 54, 4065–4078 (2015). https://doi.org/10.1007/s00526-015-0931-1

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  • DOI: https://doi.org/10.1007/s00526-015-0931-1

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