GLOBAL EXISTENCE FOR SOME STRONGLY COUPLED REACTION-DIFFUSION SYSTEMS NON-DISSIPATIVE VIA INVARIANT REGIONS TECHNIQUES.

In this work, in order to prove global existence in time of solutions for some strongly coupled reaction-di¤usion systems non-dissipative with a full di¤usion matrix, we present some simple techniques based on the construction of invariant regions. Applications to some chemical, biological and dynamics of populations models are presented. For such systems, namely when they are described by more then two equations and with nonlinearities growth more than exponential, we have not seen substantial research results in the front of global existence in time until recently. An inherent di¢ culty in systems of the type considered, is that the asymptotic sign condition in vector version (*) u . f(u) ≤ 0, for all u ϵ R m : lul ≥ C , (where C is a positive constant and f is the nonlinear term, representing the reaction), is not satis…ed. Usually this condition plays a key role in the dissipation process, and thus if it is satisfied, often leads to the existence of a bounded invariant regions and then global existence of the solutions for the system. In the case of systems formed with two equations and under the condition (*) satisfied for all u = (u 1, u 2) ϵ R 2 (i.e. C = 0) in the scalar version (i.e. u 1 :f 1 (u) and u 1 :f 2 (u) don't change sign, where f = (f 1 ; f 2)), many partial results have been obtained when the balance law is strict and recently for more general nonlinearities. In this work, in order to construct bounded invariant regions and obtain global existence, we shall alleviate the condition (*) to be satis…ed in the direction of each eigenvector (with su¢ ciently large norm) associated to the di¤usion matrix. This means that we don't impose to the reactions to satisfy the condition (*) in both the vector and the scalar versions and not necessary on the whole space R m. In addition we don't impose any conditions on the growth of the nonlinearities