Abstract functional-differential equations and reaction-diffusion systems

Authors:
R. H. Martin and H. L. Smith

Journal:
Trans. Amer. Math. Soc. **321** (1990), 1-44

MSC:
Primary 35R10; Secondary 34K30, 35K57

MathSciNet review:
967316

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a Banach space setting. The ideas are applied to reaction-diffusion systems that have time delays in the nonlinear reaction terms. The techniques presented here include differential inequalities, invariant sets, and Lyapunov functions, and therefore they provide for a wide range of applicability. The results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays.

**[1]**W. E. Fitzgibbon,*Semilinear functional differential equations in Banach space*, J. Differential Equations**29**(1978), no. 1, 1–14. MR**0492663****[2]**Avner Friedman,*Partial differential equations*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR**0445088****[3]**Jerome A. Goldstein,*Semigroups of linear operators and applications*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR**790497****[4]**Karl Kunisch and Wilhelm Schappacher,*Order-preserving evolution operators of functional differential equations*, Boll. Un. Mat. Ital. B (5)**16**(1979), no. 2, 480–500 (English, with Italian summary). MR**546470****[5]**V. Lakshmikantham and S. Leela,*Nonlinear differential equations in abstract spaces*, International Series in Nonlinear Mathematics: Theory, Methods and Applications, vol. 2, Pergamon Press, Oxford-New York, 1981. MR**616449****[6]**S. Leela and Vinicio Moauro,*Existence of solutions in a closed set for delay differential equations in Banach spaces*, Nonlinear Anal.**2**(1978), no. 1, 47–58. MR**512653**, 10.1016/0362-546X(78)90040-8**[7]**James H. Lightbourne III,*Function space flow invariance for functional-differential equations of retarded type*, Proc. Amer. Math. Soc.**77**(1979), no. 1, 91–98. MR**539637**, 10.1090/S0002-9939-1979-0539637-7**[8]**James H. Lightbourne III,*Nonlinear retarded perturbation of a linear evolution system*, Integral and functional differential equations (Proc. Conf., West Virginia Univ., Morgantown, W. Va., 1979) Lecture Notes in Pure and Appl. Math., vol. 67, Dekker, New York, 1981, pp. 201–212. MR**617050****[9]**Robert H. Martin Jr.,*Nonlinear operators and differential equations in Banach spaces*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. Pure and Applied Mathematics. MR**0492671****[10]**Robert H. Martin Jr.,*Nonlinear perturbations of linear evolution systems*, J. Math. Soc. Japan**29**(1977), no. 2, 233–252. MR**0447735****[11]**Robert H. Martin Jr.,*Asymptotic stability and critical points for nonlinear quasimonotone parabolic systems*, J. Differential Equations**30**(1978), no. 3, 391–423. MR**521861**, 10.1016/0022-0396(78)90008-6**[12]**Robert H. Martin Jr.,*A maximum principle for semilinear parabolic systems*, Proc. Amer. Math. Soc.**74**(1979), no. 1, 66–70. MR**521875**, 10.1090/S0002-9939-1979-0521875-0**[13]**R. H. Martin Jr.,*Asymptotic behavior of solutions to a class of quasimonotone functional-differential equations*, Abstract Cauchy problems and functional differential equations (Proc. Workshop, Leibnitz, 1979) Res. Notes in Math., vol. 48, Pitman, Boston, Mass.-London, 1981, pp. 91–111. MR**617214****[14]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****[15]**Samuel M. Rankin III,*Existence and asymptotic behavior of a functional-differential equation in Banach space*, J. Math. Anal. Appl.**88**(1982), no. 2, 531–542. MR**667076**, 10.1016/0022-247X(82)90211-6**[16]**George Seifert,*Positively invariant closed sets for systems of delay differential equations*, J. Differential Equations**22**(1976), no. 2, 292–304. MR**0427781****[17]**Hal Smith,*Monotone semiflows generated by functional-differential equations*, J. Differential Equations**66**(1987), no. 3, 420–442. MR**876806**, 10.1016/0022-0396(87)90027-1**[18]**Joel Smoller,*Shock waves and reaction-diffusion equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR**688146****[19]**C. C. Travis and G. F. Webb,*Existence and stability for partial functional differential equations*, Trans. Amer. Math. Soc.**200**(1974), 395–418. MR**0382808**, 10.1090/S0002-9947-1974-0382808-3**[20]**B. Z. Vulikh,*Introduction to the theory of partially ordered spaces*, Translated from the Russian by Leo F. Boron, with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki, Wolters-Noordhoff Scientific Publications, Ltd., Groningen, 1967. MR**0224522****[21]**G. F. Webb,*Asymptotic stability for abstract nonlinear functional differential equations*, Proc. Amer. Math. Soc.**54**(1976), 225–230. MR**0402237**, 10.1090/S0002-9939-1976-0402237-0

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35R10,
34K30,
35K57

Retrieve articles in all journals with MSC: 35R10, 34K30, 35K57

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-0967316-X

Keywords:
Semilinear functional differential equations,
reaction-diffusion-delay systems,
invariant sets,
differential inequalities

Article copyright:
© Copyright 1990
American Mathematical Society