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

Abstract

This paper concerns with the heat equation in the half-space ℝ+n with nonlinearity and singular potential on the boundary ∂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 Lp-spaces in which we prove linear estimates for some boundary operators arising from the Duhamel integral formulation in ℝ+n. Moreover, we investigate qualitative properties of solutions like self-similarity, positivity and symmetry around the axis Oxn⃗.