×

Existence of periodic solutions of semilinear parabolic equations and the method of upper and lower solutions. (English) Zbl 0542.35044

In this paper the authors study the existence of periodic solutions of semilinear parabolic equations with homogeneous Neumann boundary conditions. For simplicity they consider the problem in one space variable: \(u_ t-u_{xx}=f(t,x,u,u_ x),\quad u(0,x)=u(2\pi,x), u_ x(t,0)=u_ x(t,1)=0.\) Under some conditions for f, with methods used in other papers of the same authors, the existence of a solution u such that \(\alpha(t,x)\leq u(t,x)\leq \beta(t,x)\) and \(| u_ x(t,x)| \leq N\) (where N depends only on \(\alpha\), \(\beta\) and a function of a Nagumo condition) is proved.
Reviewer: I.Onciulescu

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B10 Periodic solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amann, H., Periodic Solutions of Semilinear Parabolic Equations, Nonlinear Analysis, ((1978), Academic Press: Academic Press New York), 1-29
[2] Bange, D. W., Periodic solutions of a quasilinear parabolic differential equation, J. Differential Equations, 17, 61-72 (1975) · Zbl 0291.35051
[3] Browder, F. E., Periodic solutions of nonlinear equations of evolution in infinite dimansional spaces, (Aziz, A. K., Lectures in Differential Equations, Vol. 1 (1969), Van Nostrand-Reinhold: Van Nostrand-Reinhold Princeton, N. J) · Zbl 0188.15602
[4] L. Cesari and R. KannanAccad. Sci. Milan; L. Cesari and R. KannanAccad. Sci. Milan · Zbl 0593.35008
[5] A. Castro and A. C. LazerUn. Mat. Ital.; A. Castro and A. C. LazerUn. Mat. Ital. · Zbl 0501.35005
[6] Fife, P., Solutions of parabolic boundary problems existing for all times, Arch. Rational Mech. Anal., 16, 155-186 (1964) · Zbl 0173.38204
[7] Gaines, R.; Walter, W., Periodic solutions to nonlinear parabolic differential equations, Rocky Mountain J. Math., 7, 297-312 (1977) · Zbl 0366.35006
[8] Kolesov, Ju. S., Certain tests for the existence of stable periodic soluions of quasilinear parabolic equations, Soviet Math. Dokl., 5, 1118-1120 (1964) · Zbl 0137.29502
[9] Kolesov, Ju. S., A test for the existence of periodic solutions to arabolic equations, Soviet Math. Dokl., 7, 1318-1320 (1966) · Zbl 0155.16703
[10] Kolesov, Ju. S., Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow Math. Soc., 21, 114-146 (1970) · Zbl 0226.35040
[11] Kannan, R.; Lakshmikantham, V., Periodic solutions of nonlinear boundary value problems, Nonlinear Anal., 6, 1-10 (1982) · Zbl 0494.34029
[12] R. Kannan and V. Lakshmikantham; R. Kannan and V. Lakshmikantham · Zbl 0532.34030
[13] O. A. Ladyzenskaja, V. A. Solornikov, and N. N. Ural’cevain; O. A. Ladyzenskaja, V. A. Solornikov, and N. N. Ural’cevain
[14] Lakshmikantham, V.; Leela, S., Existence and monotone method for periodic solutions of first order differential equations, J. Math. Anal. Appl., 91, 237-243 (1983) · Zbl 0525.34031
[15] Prodi, G., Soluzioni periodiche di equazioni alle derivate parziali di tipo parabolico e non lineari, Riv. Mat. Univ. Parma, 3, 265-290 (1952) · Zbl 0049.07502
[16] Seidman, T. I., Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19, 242-257 (1975) · Zbl 0281.35005
[17] Ton, B. A., Periodic solutions of nonlinear evolutions in Banach spaces, Canad. J. Math., 23, 189-196 (1971) · Zbl 0216.42302
[18] Tsai, L. Y., Periodic solutions of nonlinear parabolic differential equations, Bull. Inst. Math. Acad. Sinica, 5, no. 2, 219-247 (1977) · Zbl 0375.35031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.