Wellposedness of Second Order Evolution Equations

In this paper we study a mild wellposedness notion of a second-order abstract differential equation defined in the whole line. We establish some characterizations of this property. In the first one, we define fractional spaces that contain the solution, and we extend it continuously to the natural solution space. The second one is obtained by means of Fourier multipliers over weighted Sobolev spaces on the real line. Further, using an operator-valued version of Miklhin Fourier multipliers theorem, we exhibit some examples of operators for which the mild wellposedness is satisfied. Our results extend some previous results about abstract second order evolution equations.