2019/01/10 by Hwang, Jaeho
#35F31 #35K57 #35K91 #39A12 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1901.03038
In this paper, we consider discrete p-Laplacian parabolic equations with q-reaction term under the mixed boundary condition and the initial condition as follows: \begincases ut(x,t) = Δp,ω u(x,t) +λ\vert u(x,t) \vertq-1 u(x,t), amp;(x,t) ∈ S × (0,∞),
μ(z)\frac∂ u∂p n(z)+σ(z)\vert u(z)\vertp-2u(z)=0, amp;(x,t) ∈ ∂ S × [0,∞),
u(x,0) = u0(x) ≥ 0, amp;x ∈ S. \endcases where p>1, q>0, λ>0 and μ,σ are nonnegative functions on the boundary ∂ S of a network S, with μ(z)+σ(z)>0, z∈∂ S. Here, Δp,ω and \frac∂ ϕ∂p n denote the discrete p-Laplace operator and the p-normal derivative, respectively. The parameters p>1 and q>0 are completely characterized to see when the solution blows up, vanishes, or exists globally. Indeed, the blow-up rates when blow-up does occur are derived. Also, we give some numerical illustrations which explain the main results.