Comptes Rendus
no. 12
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]
Jean Mawhin1 ; Rafael Ortega2 ; Aureliano M. Robles-Pérez2
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
Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1089-1094.

On démontre un principe du maximum pour les solutions faibles u∈L∞(ℝ×𝕋3) de l'équation des télégraphistes utt−Δxu+cut+λu=f(t,x) en dimension spatiale trois lorsque c>0, λ∈(0,c2/4] et f∈L∞(ℝ×𝕋3) (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 utt−Δxu+cut=F(t,x,u).

A maximum principle is proved for the weak solutions u∈L∞(ℝ×𝕋3) of the telegraph equation utt−Δxu+cut+λu=f(t,x), in space dimension three, when c>0, λ∈(0,c2/4] and f∈L∞(ℝ×𝕋3) (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 utt−Δxu+cut=F(t,x,u).

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Accepté le :
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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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[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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  • Yuliia Andreieva; Kateryna Buryachenko Qualitative analysis of fourth-order hyperbolic equations, Frontiers in Applied Mathematics and Statistics, Volume 10 (2024) | DOI:10.3389/fams.2024.1467199
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  • N. Mavinga; M. N. Nkashama Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 74 (2011) no. 15, pp. 5171-5188 | DOI:10.1016/j.na.2011.05.012 | Zbl:1223.35205
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  • Jean Mawhin; Rafael Ortega; Aureliano M. Robles-Pérez Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications, Journal of Differential Equations, Volume 208 (2005) no. 1, pp. 42-63 | DOI:10.1016/j.jde.2003.11.003 | Zbl:1082.35040

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