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

This paper investigates the properties of nonnegative solutions of a quasilinear degenerate parabolic system View the MathML source{ut−div(|∇u|p−2∇u)=a∫Ωvα(x,t)dx,vt−div(|∇v|q−2∇v)=b∫Ωuβ(x,t)dx Turn MathJax on with zero Dirichlet boundary conditions in a smooth bounded domain View the MathML sourceΩ⊂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 View the MathML source∫Ωϕp−1dx∫Ωψq−1dx≤1/(ab), while if View the MathML source∫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 View the MathML source−div(|∇ϕ|p−2∇ϕ)=1 in ΩΩ, ϕ(x)|∂Ω=0ϕ(x)|∂Ω=0 and View the MathML source−div(|∇ψ|q−2∇ψ)=1 in ΩΩ, ψ(x)|∂Ω=0ψ(x)|∂Ω=0, respectively.