A New Approach for Non-Autonomous Second-Order Abstract Cauchy Problems

In this paper, well-posedness for non-autonomous second-order abstract Cauchy problems is investigated by a quasi-semigroup approach. This new approach completes the existing methods which are complex and require rigid conditions. The quasi-semigroup approach reduces the hyperbolic and parabolic conditions. The validity of this approach is confirmed by its equivalence to the fundamental solution. In the implementation, the non-autonomous second-order Cauchy problem is reduced to the first-order one, and the well-posedness follows the solution of the later problem in quasi-semigroup terms. As applications, the one-dimensional non-autonomous wave equations are considered.