Flow invariance for semilinear evolution equations under generalized dissipativity conditions

Let XX be a real Banach space, let A:D(A)⊂X→XA:D(A)⊂X→X be a linear operator which is the infinitesimal generator of a (C0)(C0)-semigroup and let B:D⊂X→XB:D⊂X→X be a nonlinear perturbation which is continuous on level sets of DD with respect to a lower semicontinuous (l.s.c.) functional φφ. We discuss the existence of a nonlinear semigroup SS providing mild solutions to the semilinear abstract Cauchy problem (SP;x)u′(t)=(A+B)u(t),t>0;u(0)=x∈D and satisfying a certain Lipschitz-like estimation and an exponential growth condition. Using the discrete schemes approximation, it is proved that the combination of a subtangential condition and a semilinear stability condition in terms of a metric-like functional is necessary and sufficient for the generation of such a semigroup SS.