Characterization of nonlinearly perturbed semigroups. (English) Zbl 0819.47081

Komatsu, Hikosaburo (ed.), Functional analysis and related topics, 1991. Proceedings of the international conference in memory of Professor Kôsaku Yosida held at RIMS, Kyoto University, Japan, July 29-Aug. 2, 1991. Berlin: Springer-Verlag. Lect. Notes Math. 1540, 85-102 (1993).
The authors use nonlinear semigroup theory to prove some results concerning semilinear evolution equations of the form \[ u'(t)= (A+ B) u(t),\qquad t> 0,\tag{1} \] where \(A\) is the infinitesimal generator of a linear \((C_ 0)\)-semigroup in a Banach space \(X\) and \(B: D\to X\) is a continuous nonlinear operator, \(D\subset X\), \(D\) closed. Motivated by the applications and assuming that \(A+B\) is quasidissipative, the main purpose is to establish necessary and sufficient conditions on the operator \(A+ B\), for the mild solutions of (1) to exist in a global sense, when \(B\) is not necessarily quasidissipative and \(D\) is not necessarily convex. Also, some results about uniqueness of mild solutions and their approximations are presented.
For the entire collection see [Zbl 0782.00076].


47H20 Semigroups of nonlinear operators
47J05 Equations involving nonlinear operators (general)
47H06 Nonlinear accretive operators, dissipative operators, etc.