Global Existence and Nonexistence for a Strongly Coupled Parabolic System with Nonlinear Boundary Conditions

This paper deals with the strongly coupled parabolic system ut = v^m△u, vt = u^n△v, （x, t） ∈Ω × （0,T） subject to nonlinear boundary conditions 偏du/偏dη = u^αv^p, 偏du/偏dη= u^qv^β, （x, t） ∈ 偏dΩ × （0, T）, where Ω 包含 RN is a bounded domain, m, n are positive constants and α,β, p, q are nonnegative constants. Global existence and nonexistence of the positive solution of the above problem are studied and a new criterion is established. It is proved that the positive solution of the above problem exists globally if and only if α 〈 1,β 〈 1 and （m ＋p）（n ＋ q） ≤ （1 - α）（1 -β）.

Global Existence and Nonexistence for a Strongly Coupled Parabolic System with Nonlinear Boundary Conditions

This paper deals with the strongly coupled parabolic system u _t = v ^m Δu, v _t = u ⁿ Δv, (x, t) ∈ Ω × (0, T) subject to nonlinear boundary conditions ∂u/∂η = u ^α v ^p , ∂v/∂η = u ^q v ^β , (x, t) ∈ ∂Ω × (0, T), where Ω ⊂ R ^N is a bounded domain, m, n are positive constants and α, β, p, q are nonnegative constants. Global existence and nonexistence of the positive solution of the above problem are studied and a new criterion is established. It is proved that the positive solution of the above problem exists globally if and only if α < 1, β < 1 and (m+ p)(n + q) ≤ (1 − α)(1 − β). Research supported by the Subsidized Scheme of Postdoctoral Research of Jiangsu Province and the Natural Science Foundation of Yancheng Normal Institute, it is also supported by the Research Scheme of the Natural Science of the Universities of Jiangsu Province (05KJB110144 and 05 KJB110063)