Heat equation with dynamical boundary conditions of reactive–diffusive type

This paper deals with the heat equation posed in a bounded regular domain Ω   of RNRN (N⩾2N⩾2) coupled with a dynamical boundary condition of reactive–diffusive type. In particular we study the problem{ut−Δu=0in (0,∞)×Ω,ut=kuν+lΔΓuon (0,∞)×Γ,u(0,x)=u0(x)on Γ, where u=u(t,x)u=u(t,x), t⩾0t⩾0, x∈Ωx∈Ω, Γ=∂ΩΓ=∂Ω, Δ=ΔxΔ=Δx denotes the Laplacian operator with respect to the space variable, while ΔΓΔΓ denotes the Laplace–Beltrami operator on Γ, ν is the outward normal to Ω, and k and l   are given real constants, l>0l>0. Well-posedness is proved for data u0∈H1(Ω)u0∈H1(Ω) such that u0|Γ∈H1(Γ)u0|Γ∈H1(Γ). We also study higher regularity of the solution.