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Existence of global coexistence state for periodic competition diffusion systems. (English) Zbl 0779.35058

Consider the following system of parabolic equations:

u t =P(u)+u[a(t,x)-b(t,x)u-c(t,x)v],v t =Q(u)+v[d(t,x)-e(t,x)u-f(t,x)v](1)

where a,,f:×Ω ¯ are positive continuous functions, which are periodic in the time t with period T>0; Ω is a bounded domain in n whose boundary Ω belongs to C 2 and P, Q are uniformly elliptic operators in ×Ω ¯, whose coefficients are T-periodic in the first variable.

When a,,f and Ω are sufficiently smooth, the author studies the existence and uniqueness of a positive solution (u,v) (u>0, v>0) of (1) such that (2) u(t+T,x)=u(t,x), v(t+T,x)=v(t,x), (3) u/ν=v/ν=0 in ×Ω, where /ν denotes differentiation in the direction of the outer normal to Ω.

The author also studies the asymptotic behavior of positive solutions (u,v) of (1), which satisfy the following initial boundary value problem:


for sufficiently smooth nonnegative functions φ, Ψ:Ω ¯ such that φ/ν=Ψ/ν=0, in Ω. To be precise, let us define from now on: B=b/a, C=c/a, E=e/d, F=f/d, and for all bounded functions g:X (some nonempty set X); g L =inf{g(x): xX} and g M =sup{g(x): xX}. In the “stable” case (6) B L >E M and F L >C M , the author proves the existence of a positive solution to (1)–(3) and he obtains a priori bounds for the positive solutions to this problem, whose components belong to C 2,1 (×Ω)C 1,0 (×Ω ¯).

Further, if (6) and (7) C M E M (F M -C L )(B M -E L )<B L F L (F L -C M )(B L -E M ) hold, then the problem (1)- -(3) has exactly one positive solution. Moreover, this solution is globally asymptotically stable.

35K57Reaction-diffusion equations
35B40Asymptotic behavior of solutions of PDE
35B35Stability of solutions of PDE
35D05Existence of generalized solutions of PDE (MSC2000)