Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications |
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| Authors: | Jean Mawhin,Aureliano M. Robles-Pé rez |
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| Affiliation: | a Institut de Mathématique Pure et Appliquée, Université Catholique de Louvain, 1348-Louvain-la-Neuve, Belgium
b Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain |
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| Abstract: | A maximum principle is proved for the weak solutions  of the telegraph equation in space dimension three utt−Δxu+cut+λu=f(t,x), when c>0, λ∈(0,c2/4] and  (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). Also, they can be employed in the study of almost periodic solutions of the forced sine-Gordon equation. A counterexample for the maximum principle in dimension four is given. |
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| Keywords: | Bounded Almost periodic Sine-Gordon |
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