Ortega, Rafael; Robles-Pérez, Aureliano M. A maximum principle for periodic solutions of the telegraph equation. (English) Zbl 0932.35016 J. Math. Anal. Appl. 221, No. 2, 625-651 (1998). This paper deals with a maximum principle for the doubly \(2\pi\)-periodic solutions of the telegraph operator defined by \[ {\mathcal L}_\lambda u= u_{tt}- u_{xx}+ cu_t-\lambda u, \] with \(c>0\). It is proved that \({\mathcal L}_\lambda\) satisfies the maximum principle if and only if \(\lambda\in [-\nu,0)\) for some \[ \nu= \nu(c)\in\Biggl({c^2\over 4},{c^2\over 4}+ {1\over 4}\Biggr]. \] Applications of this interesting principle are given to the linear telegraph operator with variable coefficients, and to a method of upper and lower solutions for the periodic solutions of nonlinear telegraph equations, which include the sine-Gordon equation. Reviewer: J.Mawhin (Louvain-La-Neuve) Cited in 3 ReviewsCited in 31 Documents MSC: 35B10 Periodic solutions to PDEs 35B50 Maximum principles in context of PDEs Keywords:doubly \(2\pi\)-periodic solutions; method of upper and lower solutions; sine-Gordon equation PDFBibTeX XMLCite \textit{R. Ortega} and \textit{A. M. Robles-Pérez}, J. Math. Anal. Appl. 221, No. 2, 625--651 (1998; Zbl 0932.35016) Full Text: DOI References: [1] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044 [2] Berestycki, H.; Nirenberg, L.; Varadhan, S. R.S., The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47, 47-92 (1994) · Zbl 0806.35129 [3] Birnir, B.; Grauer, R., An explicit description of the global attractor of the damped and driven sine-Gordon equation, Comm. Math. Phys., 162, 539-590 (1994) · Zbl 0805.35122 [4] Chow, S. N.; Hale, J. K., Methods of Bifurcation Theory (1982), Springer-Verlag: Springer-Verlag New York [5] Clément, Ph.; Peletier, A., An anti-maximum principle for second-order elliptic operators, J. Differential Equations, 34, 218-229 (1979) · Zbl 0387.35025 [6] Courant, R.; Hilbert, D., Methods of Mathematical Physics (1962), Wiley: Wiley New York · Zbl 0729.00007 [7] Dieudonné, J., Éléments d’analyse (1974), Gauthier-Villars: Gauthier-Villars Paris [8] de Figueiredo, D. G.; Mitidieri, E., Maximum principles for cooperative elliptic systems, C. R. Acad. Sci. Paris Sér. I, 310, 49-52 (1990) · Zbl 0712.35021 [9] Fucik, S.; Mawhin, J., Generalized periodic solutions of nonlinear telegraph equations, Nonlinear Anal., 2, 609-617 (1978) · Zbl 0381.35056 [10] Kim, W. S., Multiple doubly periodic solutions of semilinear dissipative hyperbolic equations, J. Math. Anal. Appl., 197, 735-748 (1996) · Zbl 0870.35069 [11] Krasnoselskii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen [12] Kreith, K.; Swanson, C. A., Kiguradze classes for characteristic initial value problems, Comp. Math. Appl., 11, 239-247 (1985) · Zbl 0607.35064 [13] Lieb, E. H.; Loss, M., Analysis (1996), American Math. Soc: American Math. Soc Providence · Zbl 0873.26002 [14] López-Gómez, J.; Molina-Meyer, M., The maximum principle for cooperative weakly coupled elliptic systems and some applications, Differential Integral Equations, 7, 383-398 (1994) · Zbl 0827.35019 [15] Mawhin, J., Periodic solutions of nonlinear telegraph equations, (Bednarek; Cesari, Dynamical Systems (1977), Academic Press: Academic Press New York), 193-210 · Zbl 0153.12603 [16] Mawhin, J., Periodic oscillations of forced pendulum-like equation, Lecture Notes in Math., 964, 458-476 (1982) [17] Peterson, A. C., On the sign of Green’s functions, J. Differential Equations, 21, 167-178 (1976) · Zbl 0292.34013 [18] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice-Hall: Prentice-Hall Englewood Cliffs · Zbl 0153.13602 [19] Rabinowitz, P. H., Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20, 145-205 (1967) · Zbl 0152.10003 [20] Vejvoda, O., Partial Differential Equations: Time-Periodic Solutions (1982), Nijhoff: Nijhoff Prague · Zbl 0183.10401 [21] Vladimirov, V. S., Generalized Functions in Mathematical Physics (1979), Mir: Mir Moscow · Zbl 0515.46033 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.