A maximum principle for bounded solutions of the telegraph equation in space dimension three
[Un principe du maximum pour les solutions bornées de l'équation des télégraphistes en dimension spatiale trois]
Comptes Rendus. Mathématique, Tome 334 (2002) no. 12, pp. 1089-1094.

On démontre un principe du maximum pour les solutions faibles uL (×𝕋 3 ) de l'équation des télégraphistes uttΔxu+cut+λu=f(t,x) en dimension spatiale trois lorsque c>0, λ∈(0,c2/4] et fL (×𝕋 3 ) (Théorème 1). Le résultat est étendu à une solution et un terme forçant appartenant à un certain espace de mesures bornées (Théorème 2). Ces résultats fournissent une méthode de sous- et sur-solutions pour l'équation semilinéaire uttΔxu+cut=F(t,x,u).

A maximum principle is proved for the weak solutions uL (×𝕋 3 ) of the telegraph equation uttΔxu+cut+λu=f(t,x), in space dimension three, when c>0, λ∈(0,c2/4] and fL (×𝕋 3 ) (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation uttΔxu+cut=F(t,x,u).

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DOI : 10.1016/S1631-073X(02)02406-8
Mawhin, Jean 1 ; Ortega, Rafael 2 ; Robles-Pérez, Aureliano M. 2

1 Département de mathématique, Université Catholique de Louvain, 1348, Louvain-la-Neuve, Belgium
2 Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain
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Mawhin, Jean; Ortega, Rafael; Robles-Pérez, Aureliano M. A maximum principle for bounded solutions of the telegraph equation in space dimension three. Comptes Rendus. Mathématique, Tome 334 (2002) no. 12, pp. 1089-1094. doi : 10.1016/S1631-073X(02)02406-8. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02406-8/

[1] Dieudonné, J. Éléments d'analyse, Tome II, Gauthier-Villars, Paris, 1974

[2] Fink, A.M. Almost Periodic Differential Equations, Lecture Notes in Math., 377, Springer, Berlin, 1974

[3] 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., Volume 251 (2000), pp. 695-709

[4] Ortega, R.; Robles-Pérez, A.M. A maximum principle for periodic solutions of the telegraph equation, J. Math. Anal. Appl., Volume 221 (1998), pp. 625-651

[5] Vladimirov, V.S. Equations of Mathematical Physics, Marcel Dekker, New York, 1971

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