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## Über dieses Buch

From the reviews: "Since E. Hille and K. Yoshida established the characterization of generators of C0 semigroups in the 1940s, semigroups of linear operators and its neighboring areas have developed into a beautiful abstract theory. Moreover, the fact that mathematically this abstract theory has many direct and important applications in partial differential equations enhances its importance as a necessary discipline in both functional analysis and differential equations. In my opinion Pazy has done an outstanding job in presenting both the abstract theory and basic applications in a clear and interesting manner. The choice and order of the material, the clarity of the proofs, and the overall presentation make this an excellent place for both researchers and students to learn about C0 semigroups." #Bulletin Applied Mathematical Sciences 4/85#1 "In spite of the other monographs on the subject, the reviewer can recommend that of Pazy as being particularly written, with a bias noticeably different from that of the other volumes. Pazy's decision to give a connected account of the applications to partial differential equations in the last two chapters was a particularly happy one, since it enables one to see what the theory can achieve much better than would the insertion of occasional examples. The chapters achieve a very nice balance between being so easy as to appear disappointing, and so sophisticated that they are incomprehensible except to the expert." #Bulletin of the London Mathematical Society#2

## Inhaltsverzeichnis

### Chapter 1. Generation and Representation

Abstract
Let X be a Banach space. A one parameter family T(t), 0 ≤t < ∞, of bounded linear operators from X into X is a semigroup of bounded linear operators on X if
(i)
T(0) = I, (I is the identity operator on X).

(ii)
T(t + s) = T, (t)T(s) for very t, s ≥ 0 (the semigroup property).

A. Pazy

### Chapter 2. Spectral Properties and Regularity

Abstract
Let T(t) be a C0 semigroup of bounded linear operators on a Banach space X. Let A be its infinitesimal generator as defined in Definition 1.1.1. We consider now the operator
$$\tilde{A}x = w - \mathop{{\lim }}\limits_{{h \downarrow 0}} \frac{{T(h)x - x}}{h}$$
(1.1)
where w — lim denotes the weak limit in X. The domain of Ã is the set of all x ϵX forr which the weak limit on the right-hand side of (1.1) exists. Since the existence of a limit implies the existence of a weak limit, it is clear that Ã extends A. That this extension is not genuine follows from Theorem 1.3 below. In the proof of this theorem we will need the following real variable results.
A. Pazy

### Chapter 3. Perturbations and Approximations

Abstract
Let X be a Banach space and let A be the infinitesimal generator of a C 0 semigroup T(t) on X, satisfying || T(t)|| ≤ Me wt . If B is a bounded linear operator on X then A + B is the infinitesimal generator of a C0semigroup S(t) on X, satisfying ||S(t)||Me(w + M || B ||)t.
A. Pazy

### Chapter 4. The Abstract Cauchy Problem

Abstract
Let X be a Banach space and let A be a linear operator from D(A)X into X. Given x ∈ X the abstract Cauchy problem for A with initial data x consists of finding a solution u(t) to the initial value problem
$$\left\{ {\begin{array}{*{20}{c}} {\frac{{du(t)}}{{dt}} = Au(t), t > 0} \hfill \\ {u(0) = x} \hfill \\ \end{array} } \right.$$
(1.1)
where by a solution we mean an X valued function u(t) such that u(t) is continuous for t ≥ 0, continuously differentiable and u(t) ∈ D(A) for t> 0 and (1.1) is satisfied. Note that since u(t) ∈ D(A) for t > 0 and u is continuous at t = 0, (1.1) cannot have a solution for x ∉ D(A).
A. Pazy

### Chapter 5. Evolution Equations

Abstract
Let X be a Banach space. For every t,0 ≤t≥ T let A(t): D(A(t))X→ X be a linear operator in X and let f (t) be an X valued function. In this chapter we will study the initial value problem
$$\left\{{_{u(s) = x.}^{\frac{{du(t)}}{{dt}} = A(t)u(t) + f(t)fors < t \leqslant T}} \right.$$
(1.1)
.
A. Pazy

### Chapter 6. Some Nonlinear Evolution Equations

Abstract
In this section we will study the following semilinear initial value problem:
$$\left\{ {\begin{array}{*{20}{c}} {\frac{{du(t)}}{{dt}} + Au(t) = f(t,u(t)), t > {{t}_{0}}} \hfill \\ {u({{t}_{0}}) = {{u}_{0}}} \hfill \\ \end{array} } \right.$$
(1.1)
where -A is the infinitesimal generator of a C0semigroup T(t), t ≥ 0, on a Banach space X and f: [t 0 ,T] X X→ X is continuous in t and satisfies a Lipschitz condition in u.
A. Pazy

### Chapter 7. Applications to Partial Differential Equations—Linear Equations

Abstract
The theory of semigroups of linear operators has applications in many branches of analysis. Such applications to Harmonic analysis, approximation theory, ergodic theory and many other subjects can be found in the general texts that are mentioned at the beginning of the bibliographical remarks.
A. Pazy

### Chapter 8. Applications to Partial Differential Equations—Nonlinear Equations

Abstract
In this section we consider a simple application of the results of Section 6.1 to the initial value problem for the following nonlinear Schrödinger equation in ∝2
$$\left\{ {_{u(x,0) = {u_0}(x)in{\mathbb{R}^2}}^{\frac{1}{i}\frac{{\partial u}}{{\partial t}} - \Delta u + k{{\left| u \right|}^2}u = 0in]0,\infty [x{\mathbb{R}^2}}} \right.$$
(1.1)
where u is a complex valued function and k a real constant. The space in which this problem will be considered is L2(R2). Defining the linear operator A 0 by D(A 0 ) = H 2 (R2)and A 0 u = — i & u for u ϵ D(A0) the initial value problem (1.1) can be rewritten as
$$\left\{{_{u(0) = {u_0}}^{\frac{{du}}{{dt}} + {A_0}u + F(u) = 0fort>0}}\right.$$
(1.2)
where F(u) = ik\u\ 2 u.
A. Pazy

### Backmatter

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