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 ) for t > 0 . By parabolic type we mean that the operator A is the infinitesimal generator of an analytic ( C 0 ) semigroup on a general Banach space X . The operator B is assumed to be locally continuous from a subset of Y into X , where Y is a Banach space which is contained in X and has a stronger norm defined through a fractional power of − A . The characterization is applied to the global solvability of the mixed problem for the complex Ginzburg–Landau equation.