Mawhin, Jean; Ortega, Rafael; Robles-Pérez, Aureliano M. Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications. (English) Zbl 1082.35040 J. Differ. Equations 208, No. 1, 42-63 (2005). The authors prove first an existence-uniqueness theorem and a maximum principle in space dimension three for the weak solutions \(u\in L^\infty(\mathbb{R}\times \mathbb{T}^3)\) of the telegraph equation \[ u_{tt}-\Delta_x u+ Cu_t+\lambda u= f(t,x)\quad\text{in }\mathbb{R}\times \mathbb{R}^3, \] where \(C> 0\), \(\lambda\in(0, C^2/4)\) and \(f\in L^\infty(\mathbb{R}\times \mathbb{T}^3)\). The result is then extended to a solution and a forcing belonging to a suitable space of bounded measures. Based on these results the authors apply the method of upper and lower solutions for the semilinear equation \[ u_{tt}-\Delta_x u+ Cu_t= F(t,x,u). \] A counterexample for the maximum principle in dimension four is also given. Reviewer: Vesa Mustonen (Oulu) Cited in 1 ReviewCited in 18 Documents MSC: 35B50 Maximum principles in context of PDEs 35L70 Second-order nonlinear hyperbolic equations 35B15 Almost and pseudo-almost periodic solutions to PDEs Keywords:Sine-Gordon; Counterexample in dimension four PDFBibTeX XMLCite \textit{J. Mawhin} et al., J. Differ. Equations 208, No. 1, 42--63 (2005; Zbl 1082.35040) Full Text: DOI References: [1] R. Dautray, J.-L. Lions, Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques, Vol. 7, Masson, Paris, 1988.; R. Dautray, J.-L. Lions, Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques, Vol. 7, Masson, Paris, 1988. [2] Dieudonné, J., Éléments d’Analyse (1974), Tome II: Tome II Gauthier-Villars, Paris [3] Fink, A. M., Almost Periodic Differential Equations. Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377 (1974), Springer: Springer Berlin · Zbl 0325.34039 [4] Mawhin, J.; Ortega, R.; Robles-Pérez, A. M., A maximum principle for bounded solutions of the telegraph equations and applications to nonlinear forcings, J. Math. Anal. Appl, 251, 695-709 (2000) · Zbl 0972.35016 [5] Mawhin, J.; Ortega, R.; Robles-Pérez, A. M., A maximum principle for bounded solutions of the telegraph equation in space dimension three, C.R. Acad. Sci. Paris, Ser. I, 334, 1089-1094 (2002) · Zbl 1002.35027 [6] Ortega, R.; Robles-Pérez, A. M., A maximum principle for periodic solutions of the telegraph equation, J. Math. Anal. Appl, 221, 625-651 (1998) · Zbl 0932.35016 [7] Szmydt, Z., Fourier Transformation and Linear Differential Equations (1980), D. Reidel Publishing Company: D. Reidel Publishing Company Dordrecht [8] Vladimirov, V. S., Generalized Functions in Mathematical Physic (1979), Mir: Mir Moscow · Zbl 0515.46034 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.