A new condition for blow-up solutions to discrete semilinear heat equations on networks

The purpose of this paper is to introduce a new condition (C)α∫0uf(s)ds≤uf(u)+βu2+γ,u>0for some α,β,γ>0 with 0<β≤α−2λ02, where λ0 is the first eigenvalue of discrete Laplacian Δω, with which we obtain blow-up solutions to discrete semilinear heat equations utx,t=Δωux,t+f(u(x,t)),x,t∈S×0,+∞,ux,t=0,x,t∈∂S×0,+∞,ux,0=u0≥0(nontrivial),x∈S¯on a discrete network S. In fact, it will be seen that the condition (C) improves the conditions known so far.