Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications

A maximum principle is proved for the weak solutions u is an element of L-infinity (R x T-3) of the telegraph equation in space dimension three u(tt) - Delta(x)u + cu(t) + lambdau = f (t, x), when c > 0, lambda is an element of (0, c(2)/4) and f is an element of L-infinity (R x 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 u(tt) - Delta(x)u + cu(t) = 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. (C) 2004 Elsevier Inc. All rights reserved.