On the heat equation with nonlinearity and singular anisotropic\n potential on the boundary

2014/08/27 by Marcelo F. de Almeida, de Almeida, Marcelo F., Lucas C. F. Ferreira +3
Mathematics · #35A01 #35B06 #35B07 #35C06 #35K05 #35K20 #42B35 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1408.6622

openalex publication_date 2014/08/27 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

This paper concerns with the heat equation in the half-space\n\ℝ₊ⁿ with nonlinearity and singular potential on the boundary\n\∂\ℝ₊ⁿ. We develop a well-posedness theory (without using\nKato and Hardy inequalities) that allows us to consider critical potentials\nwith infinite many singularities and anisotropy. Motivated by potential\nprofiles of interest, the analysis is performed in weak L^p-spaces in which\nwe prove key linear estimates for some boundary operators arising from the\nDuhamel integral formulation in \ℝ₊ⁿ. Moreover, we investigate\nqualitative properties of solutions like self-similarity, positivity and\nsymmetry around the axis overrightarrowOx_n.\n

Citations

• StrongL p -solutions of the Navier-Stokes equation inR m , with applications to weak solutions

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