Instability of periodic solutions of some evolution equations governed by time-dependent subdifferential operators (Q2266806)

This paper is concerned with the asymptotic behavior of solutions of nonlinear evolution equations of the following form in a real Hilbert space H: (E) \(u'(t)+\partial \phi^ t(u(t))\ni f(t)\), \(t\in R\), where \(\phi^ t(.)\) is a proper l.s.c. convex function on H for each \(t\in R\) and \(\partial \phi^ t\) denotes its subdifferential operator in H. The periodic case of \(\phi^ t\) with respect to t is mainly discussed, and it is shown by a simple example in 3-dimensional space that the behavior of solutions to (E) is essentially different in nature from that in the time-dependent case of \(\phi^ t\equiv \phi\). For example, even if \(\phi^ t\) is T-periodic in t and any solution of (E) is bounded in H on R, then a solution of (E) is not necessarily asymptotically T-periodic and the difference of any two T-periodic solutions is not necessarily a constant vector in H on R. However this example also suggests that any solution of (E) is asymptotically almost-periodic in H on R.