On asymptotically almost periodic mild solutions for wave equations on the whole space

Abstract

We study the existence, uniqueness and polynomial stability of forward asymptotically almost periodic (AAP-) mild solutions for the wave equation with a singular potential on the whole space $R^n$ in a framework of weak-$L^p$ spaces. First, we use a Yamazaki-type estimate for wave groups on Lorentz spaces to establish the global well-posedness of bounded mild solutions for the corresponding linear wave equations. Then, we provide a Massera-type principle which guarantees the existence of AAP-mild solutions for linear wave equations. Using the results of linear wave equations and fixed point arguments we establish the well-posedness of such solutions for semilinear wave equations. Finally, we obtain a polynomial stability for mild solutions by employing dispersive estimates.