Local Hadamard well-posedness and blow-up for reaction-diffusion equations with non-linear dynamical boundary conditions

The paper deals with local well-posedness, global existence and blow-up results for reaction-diffusion equations coupled with nonlinear dynamical boundary conditions. The typical problem studied is - ut - Δu = up-2 u in (0;∞) ; u = 0 on [0; ∞) × ωu =0 = um -2 ut on [0; ∞) 1; u (0; x) = u0 (x) in where is a bounded open regular domain of Rn (n ≥ 1), = 0 [ 1, 2 ≤ p 1+2=2, m 1 and u0 2 H1(), u0 0 = 0. After showing local well-posedness in the Hadamard sense we give global existence and blow-up results when 0 has positive surface measure. Moreover we discuss the generalization of the above mentioned results to more general problems where the terms u p -2u and ut m-2ut are respectively replaced by f (x; u) and Q(t; x; ut) under suitable assumptions on them.