Blow-up Vs.Global Finiteness For An Evolution P-Laplace System With Nonlinear Boundary Conditions

Blow-up vs. global ?niteness for an evolution p ? Laplace system with nonlinear boundary conditions In this paper we consider positive solutions of the following p?Laplacesystem: ? ? ut = div(| u |p ?2 u) + f(u,v), (x,t) ∈ ? × (0,T), (1.1) ? vt = div(| v |p ?2 v) + g(u,v), (x,t) ∈ ? × (0,T),where ? is a bounded domain in Rn with smooth boundary ??, f(·,·) 和g(·,·) are nonnegative continuous functions, nondecreasing in each variable,p > 2. With nonlinear boundary conditions ? ? ?u ? = h(u,v), (x,t) ∈ ?? × (0,T), ?η (1.2) ? ? ?v = s(u,v), (x,t) ∈ ?? × (0,T). ?ηwhere h(·,·), s(·,·) are positive C1 functions, nondecreasing in each variable.And the initial data is ? ? u(x,0) = u0(x), x ∈ ?, (1.3) ? v(x,0) = v0(x), x ∈ ?.where u0, v0 are positive, continuous functions on ?. We study the behavior of positive solutions of the problem (1.1)-(1.3)by discussing the behavior of positive solutions of the ordinary system with 5具有非线性边界条件的发展型 p ? Laplace 方程组正解的爆破性及全局有限性initial condition, the corresponding ordinary system is: ? ? ? ?(σ) = h(?(σ),ψ(σ)), ? σ ∈ R, ? ? ? ? ψ (σ) = s(?(σ),ψ(σ)), σ ∈ R, (1.4) ? ? ? ? ?(0) = ?0, ? ? ? ψ(0) = ψ0.where ?0,ψ0 are suitable nonegative constant. It is well known that the above problems have extensively applied inmany ?elds such as Chemistry, Physics, Biology and Ecology, etc. In this paper,using supersolution-subsolution method, we relate thep ? Laplace equation system (1.1)-(1.3) to the corresponding system ofnonlinear di?erential equations (1.4). By constructing a subsolution or asupersolution, we may obtain the global ?niteness or blow-up properties tothe positive solutions of the system respectively. The results we obtainedare the following:Theorem 1. If every positive solution of (1.4)blows-up, then every positivesolution of (1.1)-(1.3)blows-up. Suppose that (1.4)has global positive solutions. And suppose thatF(σ), G(σ) are monotone increasing or decreasing simultaneously, where? (σ) ψ (σ)the functions F and G are given by F(σ) = (? (σ))p ? (σ) + (? (σ))p + f(?(σ),ψ(σ)), ?2 ?1 σ ∈ R, G(σ) = (ψ (σ))p ψ (σ) + (ψ (σ))p + g(?(σ),ψ(σ)), ?2 ?1 σ ∈ R. 6吉林大学 硕士学位论文then under the above condition, we get:Theorem 2.If ∞ 1 dσ < +∞, min{? F(σ) G(σ) , } (σ) ψ (σ)then every positive solution of (1.1)-(1.3) blows-up.Theorem 3.If ∞ 1 dσ = +∞, max{? F(σ) G(σ) , } (σ) ψ (σ)then every positive solution of (1.1)-(1.3) is ?nite...