Abstract
Let X be a Banach space, let B be the generator of a continuous group in X, and let A = B 2. Assume that D( A r ) is dense in X for r an arbitrarily large positive integer and that a and b are non-negative reals. Solution representations are developed for the abstract differential equation (D 2 t + b t D t − A) · (D 2 t + a t D t − A) u(t) = 0, t > 0 corresponding to initial conditions of the form: (i) u(0+) = φ, u ( j) (0+) = 0, j = 1, 2, 3 and (ii) u 2(0+) = φ, u j (0+) = 0, j = 0, 1, 3 (with φ∈ D( A r )) for all choices of a and b.
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