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Existence Theorems for Some Weak Abstract Variable Domain Hyperbolic Problems

In this paper we prove some existence theorems for some weak problems with variable domains arising from hyperbolic equations of the type 1.1 where A = {A(t)} is, for example, a family of elliptic differential operators in space variables x = (x1, …, xn ). Thus let H be a separable Hilbert space and let V(t) ⊂ H be a family of Hilbert spaces dense in H with continuous injections i(t): V(t) → H (0 ≦ t ≦ T < ∞). Let V’ (t) be the antidual of V(t) (i.e. the space of continuous conjugate linear maps V(t) → C) and using standard identifications one writes V(t) ⊂ H ⊂ V‘(t).