An asymptotic property of solutions of wave equations
Jerome A. Goldstein · Proceedings of the American Mathematical Society · 1969
JEROME A. GOLDSTEIN Let A be a complex Hilbert space with inner product (•, •) and norm ||-||. Let A be a selfadjoint (in general unbounded) linear operator on A satisfying (1) (Ax, x) ^ 0 for all x E D(A), where D(A) denotes the domain of A. We shall consider wave equations of the form (2) u(t) + Au(t) = 0 (IER) (' =d/dt) with initial data (3) «(0) = fi E D(A), u'(0) = /2 E D(A>'2). Theorem, (i) Let A be a selfadjoint operator on X satisfying (1). Then for anyfiED(A) and for any f2ED(A112), the initial value problem (2), (3) has a unique twice strongly continuously differentiable solution. Let