Almost Periodicity of Solutions for Almost Periodic Evolution Equations
by
Zuosheng Hu
School of Mathematics and Statistics, Carleton University , Ottawa, Ontario, Canada, K1S 4K1
Coauthors: Angelo Mingarelli(Carleton University)

In this talk, we consider evolution equations in Banach space u'(t) + A(t)u(t)=f(t) where A(t):X --> X is an almost periodic operator for any t in R, and f:R --> X is an almost periodic function. It is well-known that if X=Rⁿ, A is linear and does not depend on t or if it depends on t in a periodic manner, then all bounded solutions of this equation are Bohr almost periodic, but these results cannot be extended to more general cases, for example, the case that A(t) is almost periodic, or the case that X is infinite dimensional. The main problem here is that: Under what conditions is any bounded solution of this equation almost periodic? This is a very complex question and one which has aroused the interest of many mathematicians for the years. Many authors have tried to make some progress on this question since the question was raised and some results were established. A. Haraux dealt with the case that X=R² and A is independent of t, nonlinear and monotone. The main result is that under the above assumptions, all bounded solutions of this equation are Bohr almost periodic. We deal with that case X=Rⁿ, and A(t) can depend on t in some manner. We introduce the difference-variation operator of A(t) and difference-variation equations of this equation. Using some properties of difference-variation equations, we can infer some properties of the solutions of this equation. Under some assumptions, all buonded solutions of this equation are Bohr almost periodic on R

Subject: Primary 34C27

Date received: February 12, 2003