Comptes Rendus
A maximum principle for bounded solutions of the telegraph equation in space dimension three
Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1089-1094.

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).

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).

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Published online:
DOI: 10.1016/S1631-073X(02)02406-8

Jean Mawhin 1; Rafael Ortega 2; Aureliano M. Robles-Pérez 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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Jean Mawhin; Rafael Ortega; Aureliano M. Robles-Pérez. A maximum principle for bounded solutions of the telegraph equation in space dimension three. Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1089-1094. doi : 10.1016/S1631-073X(02)02406-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02406-8/

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

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

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

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