A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

摘要

Let be a bounded linear operator in a Banach space , and let A be a closed operator in this space. Suppose that for mapping D(A) to another Banach space , and are generators of strongly continuous semigroups in . Assume finally that , where and , is a generator also. In the case where is an L (1)-type space, and is an operator of multiplication by a function , it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of with probability and is subject to the rules of with probability . We provide an approximation (a singular perturbation) of the semigroup generated by by semigroups built from those generated by and that supports this intuition. This result is motivated by a model of dynamics of Solea solea (Arino et al. in SIAM J Appl Math 60(2):408-436, 1999-2000; Banasiak and Goswami in Discrete Continuous Dyn Syst Ser A 35(2):617-635, 2015; Banasiak et al. in J Evol Equ 11:121-154, 2011, Mediterr J Math 11(2):533-559, 2014; Banasiak and Lachowicz in Methods of small parameter in mathematical biology, Birkhauser, 2014; Sanchez et al. in J Math Anal Appl 323:680-699, 2006) and is, in a sense, dual to those of Bobrowski (J Evol Equ 7(3):555-565, 2007), Bobrowski and Bogucki (Stud Math 189:287-300, 2008), where semigroups generated by convex combinations of Feller's generators were studied.