On a singular heat equation with dynamic boundary conditions
Giulio Schimperna, Antonio Segatti, Sergey Vital'evich Zelik · Asymptotic Analysis · 2016
In this paper we analyze a singular heat equation of the form [Formula: see text]. The singular term [Formula: see text] gives rise to very fast diffusion effects. The equation is settled in a smooth bounded domain [Formula: see text] and complemented with a general dynamic boundary condition of the form [Formula: see text], where [Formula: see text] is the Laplace–Beltrami operator and α and β are non-negative coefficients (in particular, the homogeneous Neumann case given by [Formula: see text] 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.