A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups
Jacek Banasiak, Adam Bobrowski · Journal of Evolution Equations · 2014
Let $${\alpha}$$ be a bounded linear operator in a Banach space $${\mathbb{X}}$$ , and let A be a closed operator in this space. Suppose that for $${\Phi_1, \Phi_2}$$ mapping D(A) to another Banach space $${\mathbb{Y}}$$ , $${A_{|{\rm ker}\, \Phi_1}}$$ and $${A_{|{\rm ker}\, \Phi_2}}$$ are generators of strongly continuous semigroups in $${\mathbb{X}}$$ . Assume finally that $${A_{|{\rm ker}\, \Phi_\text{a}}}$$ , where $${\Phi_\text{a} = \Phi_1 \alpha + \Phi_2 \beta}$$ and $${\beta = I_\mathbb{X} - \alpha}$$ , is a generator also. In the case where $${\mathbb{X}}$$ is an L 1-type space, and $${\alpha}$$ is an operator of multiplication by a function $${0 \le \alpha \le 1}$$ , it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of $${A_{|{\rm ker}\, \Phi_1}}$$ with probability $${\alpha (x)}$$ and is subject to the rules of $${A_{|{\rm ker}\, \Phi_2}}$$ with probability $${\beta (x)= 1 - \alpha (x)}$$ . We provide an approximation (a singular perturbation) of the semigroup generated by $${A_{|{\rm ker}\, \Phi_\text{a}}}$$ by semigroups built from those generated by $${A_{|{\rm ker}\, \Phi_1}}$$ and $${A_{|{\rm ker}\, \Phi_2}}$$ 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.