A condition for blow-up solutions to discrete p-Laplacian parabolic equations under the mixed boundary conditions on networks

Abstract

In this paper, we investigate the condition(Cp)α∫0uf(s)ds≤uf(u)+βup+γ,u>0 for some α> 2 , γ> 0 , and 0≤β≤(α−p)λp,0p, where p> 1 , and λp, is the first eigenvalue of the discrete p-Laplacian Δ p,ω. Using this condition, we obtain blow-up solutions to discrete p-Laplacian parabolic equations{ut(x,t)=Δp,ωu(x,t)+f(u(x,t)),(x,t)∈S×(0,+∞),μ(z)∂u∂pn(x,t)+σ(z)|u(x,t)|p−2u(x,t)=0,(x,t)∈∂S×[0,+∞),u(x,0)=u0≥0(nontrivial),x∈S, on a discrete network S, where ∂u∂pn denotes the discrete p-normal derivative. Here μ and σ are nonnegative functions on the boundary ∂S of S with μ(z) + σ(z) > 0 , z∈ ∂S. In fact, we will see that condition (Cp) improves the conditions known so far.