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

• Published in: Journal of evolution equations Vol. 15; no. 1; pp. 223 - 237
• Main Authors: Banasiak, Jacek, Bobrowski, Adam
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.03.2015
• Subjects: Analysis; Mathematics; Mathematics and Statistics; 47D06; 35B25; Singular perturbations; Boundary conditions; Semigroups of operators; 35B40; 60J15; 34G05; Convergence of semigroups
• ISSN: 1424-3199; 1424-3202; 1424-3202
• DOI: 10.1007/s00028-014-0257-z

Let α be a bounded linear operator in a Banach space X , and let A be a closed operator in this space. Suppose that for Φ 1 , Φ 2 mapping D ( A ) to another Banach space Y , A | ker Φ 1 and A | ker Φ 2 are generators of strongly continuous semigroups in X . Assume finally that A | ker Φ a , where Φ a = Φ 1 α + Φ 2 β and β = I X - α , is a generator also. In the case where X is an L 1 -type space, and α is an operator of multiplication by a function 0 ≤ α ≤ 1 , it is tempting to think of the later semigroup as describing dynamics which, while at state x , is subject to the rules of A | ker Φ 1 with probability α ( x ) and is subject to the rules of A | ker Φ 2 with probability β ( x ) = 1 - α ( x ) . We provide an approximation (a singular perturbation) of the semigroup generated by A | ker Φ a by semigroups built from those generated by A | ker Φ 1 and A | ker Φ 2 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, Birkhäuser, 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.