zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Periodic solutions of reaction diffusion systems in a half-space domain. (English) Zbl 1146.35305
Summary: This paper is concerned with the existence and stability of periodic solutions for reaction diffusion systems with nonlinear Neumann boundary conditions in a half-space domain. The approach to the problem is by the method of upper and lower solutions and the integral representation of its associated monotone iterations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iterative process in the same fashion as for parabolic initial boundary value problems. A sufficient condition for the stability of a periodic solution is given and an application is also given to a plankton allelopathic model from aquatic ecology.

35B10Periodic solutions of PDE
35K57Reaction-diffusion equations
35K50Systems of parabolic equations, boundary value problems (MSC2000)
35K60Nonlinear initial value problems for linear parabolic equations
Full Text: DOI
[1] Amann, H.: Periodic solutions of semilinear parabolic equations, , 1-29 (1978) · Zbl 0464.35050
[2] Brzychczy, S.: Monotone iterative methods for nonlinear parabolic and elliptic differential-functional equations, dissertations monographs N. 20, (1995) · Zbl 0843.35129
[3] Heikkila, S.; Lakshmikantham, V.: Monotone iterative techniques for discontinuous nonlinear differential equations, (1994)
[4] Jiang, D.; Nieto, J. J.; Zuo, W.: On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations, J. math. Anal. appl. 289, 691-699 (2004) · Zbl 1134.34322 · doi:10.1016/j.jmaa.2003.09.020
[5] Koksal, S.; Lakshmikantham, V.: Unified approach to monotone iterative technique for semilinear parabolic problems, Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 10, 539-548 (2003) · Zbl 1024.35048
[6] Kolesov, S. Ju.: Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow math. Soc. 21, 114-146 (1970) · Zbl 0226.35040
[7] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations, (1985) · Zbl 0658.35003
[8] Mukhopadhyay, A.; Chattopadhyay, J.; Tapaswi, P. K.: A delay differential equations model of plankton allelopathy, Math. biosci. 149, 167-189 (1998) · Zbl 0946.92031 · doi:10.1016/S0025-5564(98)00005-4
[9] Nieto, J. J.: An abstract monotone iterative technique, Nonlinear anal. 28, 1923-1933 (1997) · Zbl 0883.47058 · doi:10.1016/S0362-546X(97)89710-6
[10] Pao, C. V.: Nonlinear parabolic and elliptic equation, (1992) · Zbl 0777.35001
[11] Pao, C. V.: Maximum principle in differential equations, J. math. Anal. appl. 217, 129-160 (1998) · Zbl 0895.35046
[12] Pao, C. V.: Periodic solutions of parabolic systems with nonlinear boundary conditions, J. math. Anal. appl. 234, 695-716 (1999) · Zbl 0932.35111 · doi:10.1006/jmaa.1999.6412
[13] Pao, C. V.: Periodic solutions of systems of parabolic equations in unbounded domains, Nonlinear anal. 40, 523-535 (2000) · Zbl 0953.35013 · doi:10.1016/S0362-546X(00)85031-2
[14] Protter, M. H.; Weinberger, H. F.: Parabolic systems in unbounded domains: I. Existence and dynamics, (1967) · Zbl 0153.13602
[15] Song, X. Y.; Chen, L. S.: Periodic solution of a delay differential equation of plankton allelopathy, Acta math. Sci. 23A, No. 1, 8-13 (2003) · Zbl 1036.34082
[16] Tian, C. R.; Xu, F.; Liu, Y.: Stable solution of a delay parabolic equation of plankton allelopathy, J. math. Res. exposition 25, 727-733 (2005) · Zbl 1090.35508
[17] Varga, R. S.: Matrix iterative analysis, (1962)
[18] Vatsala, A. S.; Yang, J.: Monotone iterative technique for semilinear elliptic systems, Bound. value probabl., 93-106 (2005) · Zbl 1143.65388 · doi:10.1155/BVP.2005.93