A selfadjoint second order hyperbolic system

Abstract

We consider a second order hyperbolic system of the type (1) L u = u t t − B u x x = f ( x , t ) , ( x , t ) ∈ T m , where matrix B is a nonsingular constant matrix with positive eigenvalues, ( x , t ) ∈ R 2 and u , f ∈ R n . The set T m is defined to be (2) T m = { ( x , t ) | 0 ⩽ t ⩽ 1 / m , | x | ⩽ 1 − m t } , where m = min { μ k } and μ k 2 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 G n × 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 ) = ∫ ∫ T m G n × n ( x , t ; ξ , τ ) f ( ξ , τ ) d ξ d τ for any function f ∈ L 2 ( T m ) . This will imply that the operator L in (1) over the set L 2 ( T m ) 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 u t ( x , 0 ) = 0 , | x | ⩽ 1 . We further note that when B has only one eigenvalue μ 2 , 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 T μ .