Semigroups of locally Lipschitz operators associated with semilinear evolution equations of parabolic type

A characterization problem is discussed, of semigroups of locally Lipschitz operators providing mild solutions to the Cauchy problem for the semilinear evolution equation of parabolic type u′(t)=(A+B)u(t)u′(t)=(A+B)u(t) for t>0t>0. By parabolic type we mean that the operator AA is the infinitesimal generator of an analytic (C0)(C0) semigroup on a general Banach space XX. The operator BB is assumed to be locally continuous from a subset of YY into XX, where YY is a Banach space which is contained in XX and has a stronger norm defined through a fractional power of −A−A. The characterization is applied to the global solvability of the mixed problem for the complex Ginzburg–Landau equation.