We give sufficient conditions for the existence of the fundamental solution of a second order evolution equation. The proof is based on stable approximations of an operator A(t) by a sequence of bounded operators.
Let A generate a C₀-semigroup T(·) on a Banach space X such that the resolvent R(iτ,A) exists and is uniformly bounded for τ ∈ ℝ. We show that there exists a closed, possibly unbounded projection P on X commuting with T(t). Moreover, T(t)x decays exponentially as t → ∞ for x in the range of P and T(t)x exists and decays exponentially as t → -∞ for x in the kernel of P. The domain of P depends on the Fourier type of X. If R(iτ,A) is only polynomially bounded, one obtains a similar result with polynomial...
Using extrapolation spaces introduced by Da Prato-Grisvard and Nagel we prove a non-autonomous perturbation theorem for Hille-Yosida operators. The abstract result is applied to non-autonomous retarded partial differential equations.