# Co -Semigroups and Applications

**By**

- Ioan I. Vrabie, Al. I. Cuza University 700506 Iasi, Romania

The book contains a unitary and systematic presentation of both classical and very recent parts of a fundamental branch of functional analysis: linear semigroup theory with main emphasis on examples and applications. There are several specialized, but quite interesting, topics which didn't find their place into a monograph till now, mainly because they are very new. So, the book, although containing the main parts of the classical theory of Co-semigroups, as the Hille-Yosida theory, includes also several very new results, as for instance those referring to various classes of semigroups such as equicontinuous, compact, differentiable, or analytic, as well as to some nonstandard types of partial differential equations, i.e. elliptic and parabolic systems with dynamic boundary conditions, and linear or semilinear differential equations with distributed (time, spatial) measures. Moreover, some finite-dimensional-like methods for certain semilinear pseudo-parabolic, or hyperbolic equations are also disscussed. Among the most interesting applications covered are not only the standard ones concerning the Laplace equation subject to either Dirichlet, or Neumann boundary conditions, or the Wave, or Klein-Gordon equations, but also those referring to the Maxwell equations, the equations of Linear Thermoelasticity, the equations of Linear Viscoelasticity, to list only a few. Moreover, each chapter contains a set of various problems, all of them completely solved and explained in a special section at the end of the book.

The book is primarily addressed to graduate students and researchers in the field, but it would be of interest for both physicists and engineers. It should be emphasised that it is almost self-contained, requiring only a basic course in Functional Analysis and Partial Differential Equations.

### Audience

Institutes and Departments of Mathematics. Departments of Physics, Libraries of Universities.

### Book information

- Published: March 2003
- Imprint: JAI Press (Elsevier)
- ISBN: 978-0-444-51288-8

### Reviews

"The book is self-contained, requires only some acquaintance with functional analysis and partial differential equations."

Hana Petzeltova. Mathematica Bohemica, 2003.

### Table of Contents

Preface.

Chapter 1. Preliminaries.

1.1 Vector-Valued Measurable Functions.

1.2 The Bochner Integral.

1.3 Basic Function Spaces.

1.4 Functions of Bounded Variation.

1.5 Sobolev Spaces.

1.6 Unbounded Linear Operators.

1.7 Elements of Spectral Analysis.

1.8 Functional Calculus for Bounded Operators.

1.9 Functional Calculus for Unbounded Operators.

Problems.Notes.

Chapter 2. Semigroups of Linear Operators

2.1 Uniformly Continuous Semigroups.

2.2 Generators of Uniformly Continuous Semigroups.

2.3 Co-Semigroups. General Properties.

2.4 The Infinitesimal Generator.

Problems.

Notes.

Chapter 3. Generation Theorems

3.1 The Hille-Yosida Theorem. Necessity.

3.2 The Hille-Yosida Theorem. Sufficiency.

3.3 The Feller-Miyadera-Phillips Theorem.

3.4 The Lumer-Phillips Theorem.

3.5 Some Consequences.

3.6 Examples.

3.7 The Dual of a Co-Semigroup.

3.8 The Sun Dual of a Co-Semigroup.

3.9 Stone Theorem.

Problems.

Notes.

Chapter 4. Differential Operators Generating Co-Semigroups

4.1 The Laplace Operator with Dirichlet Boundary Conditions.

4.2 The Laplace Operator with Neumann Boundary Condition.

4.3 The Maxwell Operator.

4.4 The Directional Derivative.

4.5 The Schroedinger Operator.

4.6 The Wave Operator.

4.7 The Airy Operator.

4.8 The Equations of Linear Thermoelasticity.

4.9 The Equations of Linear Viscoelasticity.

Problems.

Notes.

Chapter 5. Approximation Problems and Applications

5.1 The Continuity of A → etA.

5.2 The Chernoff and Lie-Trotter Formulae.

5.3 A Perturbation Result.

5.4 The Central Limit Theorem.

5.5 Feynman Formula.

5.6 The Mean Ergodic Theorem.

Problems.

Notes.

Chapter 6. Some Special Classes of Co-Semigroups

6.1 Equicontinuous Semigroups.

6.2 Compact Semigroups.

6.3 Differentiable Semigroups.

6.4 Semigroups with Symmetric Generators.

6.5 The Linear Delay Equation.

Problems.

Notes.

Chapter 7. Analytic Semigroups.

7.1 Definition and Characterizations.

7.2 The Heat Equation.

7.3 The Stokes Equation.

7.4 A Parabolic Problem with Dynamic Boundary Conditions.

7.5 An Elliptic Problem with Dynamic Boundary Conditions.

7.6 Fractional Powers of Closed Operators.

7.7 Further Investigations in the Analytic Case.

Problems.

Notes.

Chapter 8. The Nonhomogeneous Cauchy Problem

8.1 The Cauchy Problem u'=Au+f, u(a)=&xgr;.

8.2 Smoothing Effect. The Hilbert Space Case.

8.3 Compactness of the Solution Operator from Lp(a,b;X).

8.4 The Case when (&lgr;I-A) -1 is Compact.

8.5 Compactness of the Solution Operator from Ll(a,b;X).

Problems.

Notes.

Chapter 9. Linear Evolution Problems with Measures as Data

9.1 The Problem du={Au}dt+dg, u(a)=&xgr;.

9.2 Regularity of L∞-Solutions.

9.3 A Characterization of L∞-Solutions.

9.4 Compactness of the L∞-Solution Operator.

9.5 Evolution Equations with "Spatial" Measures as Data.

Problems.

Notes.

Chapter 10. Some Nonlinear Cauchy Problems

10.1 Peano's Local Existence Theorem.

10.2 The Problem u'=f(t,u)+g(t,u).

10.3 Saturated Solutions.

10.4 The Klein-Gordon Equation.

10.5 An Application to a Problem in Mechanics.

Problems.

Notes.

Chapter 11. The Cauchy Problem for Semilinear Equations

11.1 The Problem u'=Au+f(t,u) with f Lipschitz.

11.2 The Problem u'=Au+f(t,u) with f Continuous.

11.3 Saturated Solutions.

11.4 Asymptotic Behaviour.

11.5 The Klein-Gordon Equation Revisited.

11.6 A Parabolic Semilinear Equation.

Problems.

Notes.

Chapter 12. Semilinear Equations Involving Measures

12.1 The Problem du={Au}dt+dgu with u↠ gu Lipschitz.

12.2 The Problem du={Au}dt+dgu with u↠ gu Continuous.

12.3 Saturated L∞-Solutions.

12.4 The Case of Spatial Measures.

12.5 Two Examples.

12.6 One More Example.

Problems.

Notes.

Appendix A. Compactness Results

A.1 Compact operators.

A.2 Compactness in C([a,b]; X).

A.3 Compactness in C([a,b]; Xw).

A.4 Compactness in LP(a,b; X).

A.5 Compactness in LP(a,b; X). Continued.

A.6 The Superposition Operator.

Problems.

Notes.

Solutions.

Bibliography.

List of Symbols.

Subject Index.