Existence Theorems for Some Weak Abstract Variable Domain Hyperbolic Problems

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Abstract
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).

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