B-Bounded Semigroups, Existence Families and Implicit Evolution Equations

Abstract

Let us consider a standard abstract Cauchy problem in a Banach space X:

$$\frac{{du}}{{dt}} = Au, \mathop{{\lim }}\limits_{{t \to {{0}^{ + }}}} u(t) = \mathop{u}\limits^{ \circ } . $$

(1.1)

$$ \mathop {\lim }\limits_{t \to {0^ + }} u\left( t \right) = {\text{ }}\mathop u\limits^ \circ . $$

Very often the existence of the semigroup (exp(tA)) t≥0 describing the evolution of the system is established in a non-constructive way. This is especially the case when the positivity methods are employed, see e.g. [1]. Then, very little quantitative information on the evolution is available. On the other hand, there may exist an operator B such that t→ Be ^tAcan be calculated constructively yielding some information about the evolution (note the similarity of this reasoning with that leading to C-semigroups, e.g.[9]; the final result is, however, different, see Section 5). An interesting example of this type, pertaining to the transport equation with multiplying boundary conditions, was analysed in [13] and has prompted one of the authors to define a class of evolution families which behave well if looked at through the “lens” of another operator. Such families, called B-bounded semigroups, have been introduced in [6], and analysed and applied to various problems in a few papers [2, 3, 7, 8]. In this paper we present some developments of the theory, discuss the relations between B-bounded semigroups and C-existence families, and also sketch applicability of B-bounded semigroups to solving implicit evolution equations.