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

Abstract

=forte R, and f(t+ T)=f(t) fora.e.t eR), we consider the equation (E) u’(t)+3t(u(t)) f(t), t e J, where J is an interval in R, u’(t)=(d/dt)u(t) and * isthesubdifferential of. For related studies on (E) we refer to [2, 4, 7, 8, 11, 12, 13]. In [3], Baillon and Haraux treated he time-independent case of i.e.’, and proved that any solution on J=[t0,) is asymptotically Tperiodic in the weak topology ofH and the difference of any two T-periodic solutions is a constant vector on R. Subsequently, Haraux [5] and Ishii [6] discussed the equation from the same viewpoint as in [3], when and f is almost periodic on R. In this paper we shall show by a simple example in 3-dimensional space that the equation with the time-dependent *