On a singular heat equation with dynamic boundary conditions

Article type: Research Article

Authors: Schimperna, Giulio^{a; *} | Segatti, Antonio^a | Zelik, Sergey^b

Affiliations: [a] Dipartimento di Matematica, Università di Pavia, Pavia, Italy. E-mails: giusch04@unipv.it, antonio.segatti@unipv.it | [b] Department of Mathematics, University of Surrey, Guildford, UK. E-mail: S.Zelik@surrey.ac.uk

Correspondence: [*] Corresponding author: Giulio Schimperna, Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy. E-mail: giusch04@unipv.it.

Abstract: In this paper we analyze a singular heat equation of the form ϑt+Δϑ−1=f. The singular term ϑ−1 gives rise to very fast diffusion effects. The equation is settled in a smooth bounded domain Ω⊂R3 and complemented with a general dynamic boundary condition of the form αϑt−βΔΓϑ=∂nϑ−1, where ΔΓ 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.

Keywords: very fast diffusion, Moser iterations, dynamic boundary conditions

DOI: 10.3233/ASY-151342

Journal: Asymptotic Analysis, vol. 97, no. 1-2, pp. 27-59, 2016