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Potential Analysis

, Volume 46, Issue 3, pp 589–608 | Cite as

On the Heat Equation with Nonlinearity and Singular Anisotropic Potential on the Boundary

  • Marcelo F. de Almeida
  • Lucas C. F. Ferreira
  • Juliana C. Precioso
Article

Abstract

This paper concerns with the heat equation in the half-space \(\mathbb {R}_{+}^{n}\) with nonlinearity and singular potential on the boundary \(\partial \mathbb {R}_{+}^{n}\). We show a well-posedness result that allows us to consider critical potentials with infinite many singularities and anisotropy. Motivated by potential profiles of interest, the analysis is performed in weak L p -spaces in which we prove linear estimates for some boundary operators arising from the Duhamel integral formulation in \(\mathbb {R}_{+}^{n}\). Moreover, we investigate qualitative properties of solutions like self-similarity, positivity and symmetry around the axis \(\overrightarrow {Ox_{n}}\).

Keywords

Heat equation Singular potentials Nonlinear boundary conditions Self-similarity Symmetry Lorentz spaces 

Mathematics Subject Classification (2010)

35K05 35A01 35K20 35B06 35B07 35C06 42B35 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Marcelo F. de Almeida
    • 1
  • Lucas C. F. Ferreira
    • 2
  • Juliana C. Precioso
    • 3
  1. 1.Departamento de MatemáticaUniversidade Federal de SergipeAracajuBrazil
  2. 2.Departamento de MatemáticaUniversidade Estadual de CampinasCampinasBrazil
  3. 3.Departamento de MatemáticaUnesp-IBILCESão José do Rio PretoBrazil

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