Global existence and blow-up of solutions to a nonlocal quasilinear degenerate parabolic system

This paper investigates the properties of nonnegative solutions of a quasilinear degenerate parabolic system {ut−div(|∇u|p−2∇u)=a∫Ωvα(x,t)dx,vt−div(|∇v|q−2∇v)=b∫Ωuβ(x,t)dx with zero Dirichlet boundary conditions in a smooth bounded domain Ω⊂RN(N≥1), where p,q>2p,q>2, α,β≥1α,β≥1, and a,b>0a,b>0 are positive constants. Under appropriate hypotheses, we first establish the local existence and uniqueness of solutions, then we show that whether or not the solution blows up in finite time depends on the initial data and the relations between αβαβ and (p−1)(q−1)(p−1)(q−1). In the special case of α=q−1α=q−1 and β=p−1β=p−1, we conclude that the solution exists globally if ∫Ωϕp−1dx∫Ωψq−1dx≤1/(ab), while if ∫Ωϕp−1dx∫Ωψq−1dx>1/(ab) then the solution blows up in finite time. Here ϕ(x)ϕ(x) and ψ(x)ψ(x) denote the unique solution of the following elliptic problem −div(|∇ϕ|p−2∇ϕ)=1 in ΩΩ, ϕ(x)|∂Ω=0ϕ(x)|∂Ω=0 and −div(|∇ψ|q−2∇ψ)=1 in ΩΩ, ψ(x)|∂Ω=0ψ(x)|∂Ω=0, respectively.