Properties Of Solutions To Two Kinds Of Parabolic Systems

This paper includes two parts.In the first part of this paper, we study the following localized problemwhere Ω (?) (?)^N is a bounded domain with smooth boundary (?)Ω, μ is the unit outward normal of Ω, and x₀ ∈ Ω is a fixed point. The parameters p₁,q₂ ≥ 0, P₂,q₁ > 0, and m,n > 0. The initial data u₀(x),v₀(x) ∈ C^2+α(Ω) are nontrivial and nonnegative functions, and satisfy compatibility conditions. In this part, we firstly give sufficient and neccessary conditions for that the solution blow-up in finite time, and then deduce sufficient and neccessary conditions for that two components of this solution blow-up simultaneously. Finally, we estimate the blow-up rates of this solution.In the second part of this paper, we consider a degenerate parabolic system with nonlocalizd termswhere Ω (?) (?)^N is bounded domain with smooth boundary (?)Ω, a,b > 0,p,q ≥ 1 and m,n > 1. In this part, we give the local existence of solutions, and then infer the sufficient conditions for the global existence and blow-up in finite time of solution.