Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications
نویسندگان
چکیده
A maximum principle is proved for the weak solutions uALNðR TÞ of the telegraph equation in space dimension three utt Dxu þ cut þ lu 1⁄4 f ðt; xÞ; when c40; lAð0; c=4 and fALNðR TÞ (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 Dxu þ cut 1⁄4 Fðt; x; uÞ: 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. r 2004 Elsevier Inc. All rights reserved.
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