On a singular heat equation with dynamic boundary conditions

Abstract

In this paper we analyze a singular heat equation of the form ϑt + �ϑ −1 = f . The singular term ϑ −1 gives rise to very fast diffusion effects. The equation is settled in a smooth bounded domain � ⊂ R 3 and complemented with a general dynamic boundary condition of the form αϑt − β� � ϑ = ∂nϑ −1 ,w here� � is the Laplace-Beltrami operator and α and β are non-negative coefficients (in particular, the homogeneous Neumann case given by α = β = 0 is included). For this problem, we first introduce a suitable weak formulation and prove a related existence result. For more regular initial data, we show that there exists at least one weak solution satisfying instantaneous regularization effects which are uniform with respect to the time variable. In this improved regularity class, uniqueness is also shown to hold.