Consider the following system of parabolic equations:
where are positive continuous functions, which are periodic in the time with period ; is a bounded domain in whose boundary belongs to and , are uniformly elliptic operators in , whose coefficients are -periodic in the first variable.
When and are sufficiently smooth, the author studies the existence and uniqueness of a positive solution , of (1) such that (2) , , (3) in , where denotes differentiation in the direction of the outer normal to .
The author also studies the asymptotic behavior of positive solutions of (1), which satisfy the following initial boundary value problem:
for sufficiently smooth nonnegative functions , such that , in . To be precise, let us define from now on: , , , , and for all bounded functions (some nonempty set ); : and : . In the “stable” case (6) and , 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 .
Further, if (6) and (7) hold, then the problem (1)- -(3) has exactly one positive solution. Moreover, this solution is globally asymptotically stable.