GLOBAL EXISTENCE FOR THE HEAT EQUATION WITH NONLINEAR DYNAMICAL BOUNDARY CONDITIONS ENZO VITILLARO Abstract. The paper deals with local and global existence for the solutions of the heat equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied is ut − Δu =0 in (0, ∞) × Ω, u =0 on [0, ∞) × Γ 0 , ∂u ∂ν = −|u t | m-2 u t + |u| p-2 u on [0, ∞) × Γ 1 , u(0,x)= u 0 (x) on Ω, where Ω ⊂ n (n ≥ 1) is a regular and bounded domain, ∂Ω=Γ 0 ∪ Γ 1 , m> 1, 2 ≤ p<r, where r = 2(n − 1)/(n − 2) when n ≥ 3, r = ∞ when n =1, 2, and u 0 ∈ H 1 (Ω), u 0 = 0 on Γ 0 . We prove local existence of the solutions in H 1 (Ω) when m > r/(r +1 − p) or n =1, 2, and global existence when p ≤ m or the initial data are inside the potential well associated to the stationary problem. 1. Introduction and main results We consider the problem (1)            u t − Δu =0 in (0, ∞) × Ω, u =0 on [0, ∞) × Γ 0 , ∂u ∂ν = −Q(t, x, u t )+ f (x, u) on [0, ∞) × Γ 1 , u(0,x)= u 0 (x) on Ω, where u = u(t, x), t ≥ 0, x ∈ Ω, Δ denotes the Laplacian operator, with respect to the x variable, Ω is a bounded open subset of R n (n ≥ 1) of class C 1 (see [4]), ∂ Ω=Γ 0 ∪ Γ 1 ,Γ 0 ∩ Γ 1 = ∅,Γ 0 and Γ 1 are measurable over ∂ Ω, endowed with the (n − 1) – dimensional Lebesgue measure λ n−1 . These properties of Ω, Γ 0 and Γ 1 will be assumed, without further comments, throughout the paper. The initial datum u 0 belongs to H 1 (Ω) and u 0 = 0 on Γ 0 . Moreover Q repre- sents a nonlinear boundary damping term, i.e. Q(t, x, v)v ≥ 0, and f represents a nonlinear source term, i.e. f (x, u)u ≥ 0. Local and global existence for the solutions of problems like (1) has been widely studied when Q ≡ 0 (parabolic problems with nonlinear boundary conditions) or Q ≡ u t (parabolic problems with dynamical boundary conditions). We respectively refer to [2], [8], [16], [17], [18] and to [6], [9], [10], [11], [14], [15]. The quoted papers contain, roughly, three different kinds of results: local exis- tence (with various regularity assumptions on u 0 and f ), global existence when f Work done in the framework of the M.U.R.S.T. project ”Metodi variazionali ed equazioni differenziali nonlineari”. 1