Global Existence And Nonexistence For A Degenerate And Strongly Coupled Parabolic System

Nonlinear diffusion equations, as an important class of partial diffusion equations, come from a variety of diffusion phenomena appeared widely in nature. They arise from many fields such as physics, chemistry and dynamics of biological groups. In recent years, the study in this direction attracts a large number of mathematicians both in China and aboard, and remarkable progress has been achieved on the local existence of classical solutions, global existence, blow-up as well as estimates on the blow-up time and blow-up set, which enrich enormously the theory of partial differential equations. Until now, the study of diffusionequations is still a very active research area.Early in 1986, A. Friedman and J. B. Mcleod studied the following type of single equationand obtained that the solution of (1) exists globally ifλ₁ >a while it blows up in finite time in case ofλ₁ < a. Moreover, when the solution blows up in finite time, the (Lebesgue) measure of the blow-up set is positive. Later, Gage considered the blow-up case again for p = 2, gave an estimate for the blow-up time and made a more detailed study of the blow-up set. For the caseλ₁> a and p > 2, in 1993, Zimmer studied the global existence of solutions to (1). For the case 1 < p < 2, Chen proved that the solution of (1) blows up in finite time for large domainΩ(meaningλ₁ < a) and large initial data. At the same time, Weigner considered Problem (1) for the case p > 1 and proved that the solution of (1) blows up in finite time ifλ₁ < a. Moreover, he obtained the uniqueness of solution when p≥2. Recently, Wang et al. discussed (1) for the case p > 1. At first, they obtained the uniqueness result, and then proved that all positive solutions of (1) exist globally if and only ifλ₁≥a. Similar problems have also been discussed by Y. Su and C. Mu et al.In 2000, Duan and Zhou investigated the following type of degenerate and quasilinear parabolic systemThey discussed Problem (2) for one dimension case and showed if max{a₁,a₂} <λ₁ and u₀xx+a₁u₀≤0, v_oxx+a₂v₀≤0 inΩ, Problem (2) has global solution, while if min{a₁, a₂} >λ₁ and u₀xx+a₁u₀≥0, v_oxx+a₂v₀≥0 inΩ, then Problem (2) has solution which blows up in finite time. These results were extended by Wang and Xie for multiple-dimension case under a little weaker assumptions on initial data.In 2006, Kwang Ik Kim and Zhigui Lin studied the following mutualistic modelThey proved that Problem (3) admits a unique global solution which is bounded when b₂c₁ < b₁c₂ and the solution blows up in finite time when the two species are strongly mutualistic, i.e., b₂c₁ > b₁c₂ andλ₁ < min{a₁,a₂}.The main purpose of this paper is to study the global existence and blow-up properties of the following problem:whereΩis a bounded domain in R^N with smooth boundary (?)Ωand a_i, b_i,c_i,(i = 1,2), p, q are positive constants. Initial datum (u₀, v₀) satisfiesHere n is the outward normal vector on (?)Ω. This model can be used to describe the change of the densities of the biological groups. In biological terms, the unknowns u and v represent the spatial densities of the species at time t and a_i is its respective net birth rate. The coefficientsb₁ and c₂ are intra-specific competitions, whereas b₂ and c₁ are those of inter-specific competitions.We say that (u, v) is a classical solution of Problem (4) if (u, v)∈[C^2,1(Ω×(0,T))∩C((?)×[0, T))]² for some 0 < T≤∞and (u, v) satisfies the differential equation inΩ×(0, T) and the initial and boundary conditions continuously.We say that (u, v) is a maximal solution of Problem (4), if it is a solution of Problem (4) and u≥(?), v≥(?) provided that (?) is a solution of Problem (4).We say that a positive solution of (4) blows up in finite time if there exists a T: 0 < T <∞such that (u,v) exists in (0, T) and lim sup_t→T max_?{u(·,t) + v(·, t)} =∞.In Section 3, we obtain the local existence theorem of classical solutions to Problem (4) by using regularized methods as well as a prior estimate, and getTheorem 1 (Local existence theorem.) Assume that (u₀(x), v₀(x)) satisfies (5), then (4) admits a local solutionfor some T : 0 < T <∞.In Section 4, we give a sufficient condition under which Problem (4) admits global solutions, and we getTheorem2 (Global existence theorem.) Assumeλ₁ >max{a₁,a₂}, b₁c₂≥b₂c₁ then Problem (4) admits a global positive solution (u, v)∈[C^2,1(Ω×(0,∞))∩C((?)×[0,∞))]². Moreover, there exists two constants K₁, K₂ > 0 such that hold for all global positive classical solutions.In the last section of this paper, we discuss the global nonexistence and blow-up propertiesof Problem (4) and the main result is the followingTheorem 3 (Global existence theorem.) Ifλ₁ < min{a₁,a₂}, c₁≥b₁, b₂≥c₂, Problem (4) has no global positive solution. Moreover, if the initial datum (u₀, v₀)∈[C⁴(?)]² and satisfiesâ–³u₀ + u₀(a₁ - b₁ u₀ + c₁v₀)≥0,â–³v₀ + v₀(a₂ + b₂u₀- c₂v₀)≥0 inΩ;â–³u₀=â–³v₀= 0 on (?)Ω,then the maximal solution of (4) blows up in finite time.