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Dissipation and asymptotic behavior of some reaction-diffusion systems. (English) Zbl 0551.92020
Differential equations and their applications, Equadiff 5, Proc. 5th Czech. Conf., Bratislava 1981, Teubner-Texte Math. 47, 119-122 (1982).
[For the entire collection see Zbl 0507.00006.]
The paper deals with a particular type of reaction-diffusion problems that are of interest when we have n species involved and only $$r\leq n$$ species for which we have to take diffusion into account. A ”mass-action kinetics” of the type linear combination of powers is considered. A particular case is a model of a certain poly-condensation process proposed by T. M. Peel and T. G. Davis, J. Polym. Sci. 11, 1671-1682 (1973).
Under certain regularity conditions, the author shows that the initial value problem has a unique solution for a finite time interval and, under further conditions (like nonnegative dissipation rate), he finds a Lyapunov function. He then applies these results to the particular case cited above, extending the existence of a unique solution to the time interval [0,$$\infty)$$ and studying the asymptotic behavior of the solution.
The results are stated without proofs or with extremely sketchy ones and no references are given to other papers where such proofs may be found. This is probably a consequence of the short number of pages the author was allowed to use. Such limitations, commonly imposed on Conference Proceedings, can make almost useless an otherwise potentially good paper.
Reviewer: C.A.Braumann

MSC:
 92Exx Chemistry 35B40 Asymptotic behavior of solutions to PDEs 92D25 Population dynamics (general) 35Q99 Partial differential equations of mathematical physics and other areas of application 35K99 Parabolic equations and parabolic systems