Solutions to systems of nonlinear reaction-diffusion equations

✍ Scribed by Gerald Rosen

• Publisher: Springer
• Year: 1975
• Tongue: English
• Weight: 641 KB
• Volume: 37
• Category: Article
• ISSN: 1522-9602
• DOI: 10.1016/s0092-8240(75)80033-4

✦ Synopsis

General criteria which either preclude time-periodic dissipative structure solutions or imply asymptotically steady solutions are derived for generic systems of reaction-diffusion equations ~ct[at = DtV2c~ + Qt(c) subject to boundary conditions of practical interest, where the enumerator index i runs 1 to n, c t = ct(x, t) denotes the concentration or density of the ith participating molecular or biological species, D~ is the diffusivity constant for the ith species, and Ql(e), an algebraic function of the n-tuple e = (cl . . . . . %), expresses the local rate of production of the ith species due to chemical reactions or biological interactions. It is demonstrated that certain functionals of c which decrease monotonically with time can often be found, as exemplified here for Volterra and Verhulst-Volterra n-species model systems, and thus time-periodic dissipative structure solutions are precluded for such systems of reactlon-diffusion equations. It is shown that all solutions to a generic system of reaction-diffusion equations evolve dynamically to a unique steady state, lim ct(x, t) = e~(x), ~-*aO if the diffusivity constants are all sufficiently large in magnitude. A necessary condition for the existence of a periodic solution (either spatially uniform or non-uniform) is formulated in terms of the curl of q(e) in e-space. Finally, necessary and sufficient conditions are derived for the existence of time-periodic dissipative structure solutions in cases of "weak diffusion" with the reaction rate terms dominant in the governing equations.

1. Introduction.

Considerable interest has been a t t a c h e d t o t h e recent experim e n t a l ( H e r s c h k o w i t z -K a u f m a n , 1970; Z h a b o t i n s k y a n d Zaikin, 1973) a n d theoretical studies (Glansdorff a n d