A selfadjoint second order hyperbolic system

We consider a second order hyperbolic system of the type Lu = utt −Buxx = f (x, t), (x, t) ∈ Tm, (1) where matrix B is a nonsingular constant matrix with positive eigenvalues, (x, t) ∈ R2 and u, f ∈ Rn. The set Tm is defined to be Tm = (x, t) 0 t 1/m, |x| 1− mt , (2) where m = min{μk} and μ2k is any eigenvalue of the matrix B. We will show that, under the condition u(x, 0) = 0, |x| 1, a symmetric Green’s function Gn×n can be constructed [K. Kreith, A selfadjoint problem for the wave equation in higher dimensions, Comput. Math. Appl. 21 (5) (1991) 12–132] so that u(x, t) = Tm Gn×n(x, t ;ξ, τ)f (ξ, τ)dξ dτ (3) for any function f ∈ L2(Tm). This will imply that the operator L in (1) over the set L2(Tm) of functions given by Eq. (3) and u(x, 0) = 0, |x| 1, is selfadjoint. We also note that the same result holds for u in (1), under the condition that ut (x, 0) = 0, |x| 1. We further note that when B has only one eigenvalue μ2, the function u in Eq. (3) satisfies a boundary condition similar to that of Kalmenov [T. Kalmenov, On the spectrum of a selfadjoint problem for the wave equation, Akad. Nauk. Kazakh SSR Vestnik 1 (1983) 63–66] on the characteristic boundaries of Tμ. © 2006 Elsevier Inc. All rights reserved