Functional evolution equations. (Q1410254)

The paper concerns the functional evolution equation \[ u'(t) \in B u + F(t, u_t) , \;\;t >0, \] \[ u_0(s)= \phi(s)\in C^1([-r,0];X), \] where \(X\) is a real Banach space, not necessarily reflexive, \(F : [0,\infty) \times C([-r,0];X) \to X \) is a single-valued function and \(B: D(B)\subset X \to X\) is a multi-valued nonlinear operator. From the abstract: ``It is shown by the method of lines combined with the Crandall-Pazy theorem, that for the initial data \(u_0\) in a generalized domain, this equation has a limit solution, which is Lipschitz continuous in \ \(t\), and that this limit solution is a unique strong one if further assumptions on \(B\) are imposed, and that the zero solution is asymptotically stable. An application is given to a class of partial differential boundary value problems''.