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

Le The Sac and Pham Truong Xuan

Mathematische Nachrichten, 2025, vol. 298, issue 8, 2675-2690

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 Rn$\mathbb {R}^n$ in a framework of weak‐Lp$L^p$ 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.

Date: 2025
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