Characterization of Nonlinear Semigroups Associated with Semilinear Evolution Equations

Abstract

Nonlinear continuous perturbations of linear dissipative operators are considered from the point of view of the nonlinear semigroup theory. A general class of nonlinear perturbations of linear contraction semigroups in a Banach space $X$ is introduced by means of a lower semicontinuous convex functional $[{\text {unk}}]:X \to [0,\infty ]$ and two notions of semilinear infinitesimal generators of the associated nonlinear semigroups are formulated. Four types of necessary and sufficient conditions are given for a semilinear operator $A + B$ of the class to be the infinitesimal generator of a nonlinear semigroup $\{ S(t):t \geqslant 0\}$ on the domain $C$ of $B$ such that for $x \in C$ the $C$-valued function $S( \cdot )x$ on $[0,\infty )$ provides a unique mild solution of the semilinear evolution equation $u’(t) = (A + B)u(t)$ satisfying a growth condition for the function $[{\text {unk]}}(u( \cdot ))$. It turns out that various types of characterizations of nonlinear semigroups associated with semilinear evolution equations are obtained and, in particular, a semilinear version of the Hille-Yosida theorem is established in a considerably general form.