A selfadjoint second order hyperbolic system

摘要

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,fRn. The set Tm is defined to be Tm={(x,t)|0≤t≤1/m,|x|≤1-mt} (2) where m=min{μk} and 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 Gnxn(x,t;ξ,τ)f(ξ,τ) dξ dτ (3)for any function fL2(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.