A selfadjoint second order hyperbolic system

AbstractWe consider a second order hyperbolic system of the type(1)Lu=utt−Buxx=f(x,t),(x,t)∈Tm, 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(2)Tm={(x,t)|0⩽t⩽1/m,|x|⩽1−mt}, where m=min{μk} and μk2 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(3)u(x,t)=∫∫TmGn×n(x,t;ξ,τ)f(ξ,τ)dξdτ 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