This 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 u_t − Δu = 0 in (0, ∞) × Ω, u = 0 on [0, ∞) × Λ_0, ∂u/∂v = −|u_t|^{m−2}u_t + |u|^{p−2}u on [0, ∞) × Λ1, u(0, x) = u_0 (x) on Ω, where Ω ⊂ R^n (n ≥ 1) is a regular and bounded domain, ∂Ω = Λ_ ∪ Λ_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 Λ_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 datum is inside the potential well associated to the stationary problem.
Global existence for the heat equation with nonlinear dynamical boundary conditions
VITILLARO, Enzo
2005
Abstract
This 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 u_t − Δu = 0 in (0, ∞) × Ω, u = 0 on [0, ∞) × Λ_0, ∂u/∂v = −|u_t|^{m−2}u_t + |u|^{p−2}u on [0, ∞) × Λ1, u(0, x) = u_0 (x) on Ω, where Ω ⊂ R^n (n ≥ 1) is a regular and bounded domain, ∂Ω = Λ_ ∪ Λ_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 Λ_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 datum is inside the potential well associated to the stationary problem.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
