A condition for blow-up solutions to discrete semilinear wave equations on networks. Issue 6 (13th April 2022) - Choi, Min-Jun
Abstract : In this paper, we consider discrete semilinear wave equations 1 u t t x, t = Δ ω u x, t + f u x, t, x, t ∈ S × 0, + ∞, u x, t = 0, x, t ∈ ∂ S × 0, + ∞, u x, 0 = u 0 x, u t x, 0 = v 0 x, x ∈ S, where S is a network with boundary ∂ S . The main contribution of this work is to introduce a condition 2 ( C ) α ∫ 0 u f s d s ≤ u f ( u ) + β u 2 + γ, u ∈ R, for some α > 2 and β, γ ≥ 0 with 0 ≤ β ≤ α − 2 λ 0 / 2, where λ 0 is the first eigenvalue of discrete Laplacian Δ ω, and we use the concavity method to obtain the blow-up solutions to the above equations.