Asymptotic Behavior Of Solutions For Systems Of Periodic Reaction-Diffusion Equations On Unbounded Domains

In this thesis, we obtain the upper and lower solutions methods about local asymptotic behavior of solutions for systems of periodic reaction diffusion equationsby using the integral representation theory of second order linear parabolic boundary-value problems and the ladder technique. Asymptotic behavior of solutions of problem (1.1) intimatedly related to upper and lower solution and the maximal and the minimal periodic solutions of the corresponding periodic steady-state problemwhere is a unbounded domains in Rn with boundary , including the whole space Rn, the exterior of a bounded domain and the half space .T is a positive constant. For each i = 1,2,....., N, if the coefficients of operator Li, the reaction function ft (t, x,) and boundary function fy (/, x) are properly smooth and T -periodic on [0,+ ) and the reaction function is quasimonotone decreasing on J, then our main results of this paper are stated as follows:If problem (1.2) exist a pair of ordered upper and lower solution tnat are bounded on , then problem (1.2)exist the maximal periodic solution UT = and the minimal periodic solution on J and for any , the solution U(t, x) = (uJ (t, x), u2 (t, x), ..., UN (t, x)) of problem (1.1) satisfiesIn particular, if U*(t,x) is unique periodic solution of problem (1.2) on J, thenwhereIn the end, we applied the above results to discuss local asymptotic behavior of a periodic competitor-competitor-mutualist on the whole Rn.The main results of this paper extend the results of C. V. Pao[4,7,8].