Gaston N'Guerekata, Morgan State University
"Almost automorphic evolution equations"
The aim of this talk is threefold: 1. We will present the method of reduction to prove the existence of almost automorphic solutions to an evolution equation in finite dimensional space. 2. We will introduce the concept of uniform spectrum of bounded functions as a tool to resolve an open problem for the existence of almost automorphic mild solutions to evolution equations of the form x'(t)=Ax(t)+f(t) in an infinite dimensionsional space where A is a (generally unbounded operator) and f an almost automorphic function. 3. We will show, using the sub and super solution method combined with the Bohr compactification of the euclidean space, that a second order elliptic almost periodic equation may not have almost periodic solutions but many almost automorphic solutions in the envelope of the equation. We apply the result to an almost periodically forced pendulum. All results are the speaker's contributions.