A condition for blow-up solutions to discrete semilinear wave equations on networks
Min-Jun Choi · Applicable Analysis · 2020
In this paper, we consider discrete semilinear wave equations uttx,t=Δωux,t+fux,t,x,t∈S×0,+∞,ux,t=0,x,t∈∂S×0,+∞,ux,0=u0x,utx,0=v0x,x∈S, where S is a network with boundary ∂S. The main contribution of this work is to introduce a condition (C)α∫0ufsds≤uf(u)+βu2+γ,u∈R, for some α>2 and β,γ≥0 with 0≤β≤α−2λ0/2, where λ0 is the first eigenvalue of discrete Laplacian Δω, and we use the concavity method to obtain the blow-up solutions to the above equations.