Cauchy Problems

Abstract

In this chapter we study systematically well-posedness of the Cauchy problem. Given a closed operator A on a Banach space X we will see in Section 3.1 that the abstract Cauchy problem

$$\{ \begin{array}{*{20}{c}}{u'(t) = Au(t)\;(t \geqslant 0)} \\ {u(0) = x} \end{array} $$

is mildly well posed (i.e., for each x ∈ X there exists a unique mild solution) if and only if the resolvent of A is a Laplace transform; and this in turn is the same as saying that A generates a C ₀-semigroup. Well-posedness in a weaker sense will lead to generators of integrated semigroups (Section 3.2). The real representation theorem from Section 2.2 will give us directly the characterization of generators of C ₀-semigroups in terms of a resolvent estimate; namely, the Hille-Yosida theorem. When the operators are not densely defined, we obtain Hille-Yosida operators which are studied in detail in Section 3.5. Also for results on approximation of semigroups in Section 3.6 we can use corresponding results on Laplace transforms from Section 1.7. Much attention is given to holomorphic semigroups which are particularly simple to characterize by means of the results of Section 2.6. We consider not only holomorphic semigroups which are strongly continuous at 0, but more general holomorphic semigroups which will be useful in applications to the heat equation with Dirichlet boundary conditions in Chapter 6. When the holomorphic semigroup exists on the right half-plane, the boundary behaviour is of special interest. If the semigroup is locally bounded, then a boundary C ₀-group is obtained on the imaginary axis. This case is particularly important for fractional powers (see also the Notes of Section 3.7) and for the second order problem (Section 3.16). When the holomorphic semigroup is polynomially bounded we obtain k-times integrated semigroups where the k depends on the degree of the polynomial. A typical example is the Gaussian semigroup. Its boundary is governed by the Schrödinger operator iΔ, which we study in Section 3.9 and in Chapter 8. The last three sections are devoted to the second order Cauchy problem; i.e., to the theory of cosine functions. A central result will be to establish a unique phase space on which the associated system is well posed. This is a particularly interesting special case of the intermediate spaces which are constructed in Section 3.10 for integrated semigroups.