Abstract Cauchy problems: three approaches (Q2715718)

Among the many books and monographs that introduce the reader to the use of methods which investigate the abstract Cauchy problem (CP) NEWLINE\[NEWLINEu'(t)=Au(t), \quad t\geq 0, \qquad u(0)=x\in X, \tag{*}NEWLINE\]NEWLINE with operators not generating \(C_0\) semigroups, the present one is unique in the following respect: It shows how the semigroup method, the abstract distribution method and the regularization method may be applied to investigate such problems. The book, which is self-contained and accessible to nonspecialists, presents the three methods above as well as their connection. A principal concept we meet in the first two methods is the relaxation of ``well-posedness'' so that a (CP) that is not well-posed in the classical sense might be well-posed in a certain other sense. In the first approach by using integrated \(C\)-regularized and \(\kappa\)-convoluted semigroups one can give solution operators whose corresponding solutions are stable with respect to norms that are stronger than the basic norm. The second method leads to the construction of a family of generalized solution operators, while the third method shows how one can get an approximate solution to an ill-posed local (CP) whose corresponding operator is a regularizing operator. In more details the book is organized as follows: NEWLINENEWLINENEWLINEChapter 0: Here, the authors want to present the illustration and motivation of some of the ideas and notions discussed in the book. They present the heat and the wave equation to show how one can formulate abstract Cauchy problems in several spaces. NEWLINENEWLINENEWLINEChapter 1: It starts with a brief summary of very basic notions and results from the theory of \(C_{0}\)-semigroups, and then offers some essential properties of an abstract (CP). Two examples are given to illustrate the semigroup methods. Section 1.2 contains a deep discussion of nondegenerate exponentially bounded and local integrated semigroups, and shows their connection with the (CP). Several examples illustrate the discussion. In section 1.3 the authors connect the existence of a \(\kappa\)-convoluted semigroup with the well-posedness of a \(\Theta\)-convoluted (CP) with the behavior of the resolvent, which in the convoluted case is allowed to increase exponentially. Section 1.4 discusses \(C\)-regularized semigroups and families of bounded operators connected with (CPs) that are not uniformly well-posed. A degenerate (CP) has the form \(Bu'(t)=Au(t), t\geq 0, u(0)=x, KerB\neq \{0\}\) (DP) where \(B,A\) are linear operators in some Banach spaces. In section 1.5 degenerate integrated semigroups and \(C_{0}\)-semigroups connected with degenerate (CPs) are investigated. In section 1.6 the authors use the technique of degenerate semigroups with multivalued generators for studying the well-posedness of the inclusion (CP) \(u'(t)\in Au(t),t\geq 0, u(0)=x, \) where \(A\) is here a multivalued operator. As a consequence, among others, a criterion is obtained for the well-posedness of the (CP), (DP). The chapter ends with section 1.7, where semigroup methods are used to study the well-posedness of abstract second-order (CPs) of the form \(u''(t)=Au'(t)+Bu(t), t\geq 0, u(0)=x, u'(0)=y\) and \(Qu''(t)=Au'(t)+Bu(t), t\geq 0, u(0)=x, u'(0)=y,\) with \(\text{Ker }Q\neq \{0\}.\)NEWLINENEWLINENEWLINEChapter 2: The main aim in this chapter is to obtain necessary and sufficient conditions for the well-posedness in the space of distributions in terms of the resolvent of the operator \(A\) for (*), where the initial value \(x\) lies in the space of distributions and exponential distributions. Well-posedness in the sense of distributions is also discussed. Moreover in this chapter a (CP) is studied with an operator \(A\) having the resolvent in a certain region \(\Lambda\) smaller than a logarithmic region. The well-posedness of such a (CP) in spaces of ultradistributions is investigated. Such spaces are the dual spaces of infinitely differentiable functions with a locally convex topology. (These spaces are more general than the space of Schwartz distributions.) It is shown that the existence of a solution operator ultradistribution is equivalent to the existence of exponential estimates for the resolvent of \(A\) in \(\Lambda.\) NEWLINENEWLINENEWLINEIn the final chapter of the book, the authors consider regularizing operators that allow the construction of an approximate solution to (*) which is stable in \(X\) with respect to the approximate parameters. In section 3.1 the ill-posed (CP) is faced by three ``differential'' regularization methods. The main method of regularization and the regularized semigroups are discussed in section 3.2 and the book ends with the regularization of ``slightly'' ill-posed problems. NEWLINENEWLINENEWLINEOverall, this book is a good exposition of the three methods in discussing abstract Cauchy problems. Besides its appeal to beginners, the material may also be of use to experienced researchers in the field. The special techniques developed here that associate the application of each method deserve wide attention.NEWLINENEWLINENEWLINEThe book contains a bibliography with 297 references, a glossary of the notations used in the book and an index of the terminology.