Evolution Equations

Abstract

We consider here, in the spirit of the previous chapters, an equation $\dot u = - A(t)u$ with a periodic unbounded operator coefficient in a Hilbert space. In section 5.1 some kind of hypoellipticity (more general than parabolicity) is assumed. We deal here with solutions defined on the whole axis (Cauchy problem is in general ill-posed for this class of equations). The next section is devoted to some degenerate cases arising in applications. We discuss Cauchy problem for abstract parabolic problems in section 5.3 and parabolic and elliptic boundary problems in cylindrical domains in section 5.4. The last section contains some remarks and references.