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2018 | Book

Unbounded Linear Operators and Evolution Equations Read chapter Read first chapter

Nonlinear Vibrations and the Wave Equation

Author: Dr. Alain Haraux

Publisher: Springer International Publishing

Book Series : SpringerBriefs in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book gathers the revised lecture notes from a seminar course offered at the Federal University of Rio de Janeiro in 1986, then in Tokyo in 1987. An additional chapter has been added to reflect more recent advances in the field.

Table of Contents

Frontmatter
Chapter 1. Unbounded Linear Operators and Evolution Equations
Abstract
In this chapter, we collect some basic tools essentially related to the linear semi-group theory, which will reveal very useful in the subsequent more specialized chapters.
Alain Haraux
Chapter 2. A Class of Abstract Wave Equations
Abstract
In this chapter, we apply the general results of Chap. 1 to define and study the solutions to some second order evolution equations of the form
$${u}'' + {Lu}({t}) = {f}({t})$$
where L is an unbounded, positive and self-adjoint operator on a real infinite-dimensional Hilbert space.
Alain Haraux
Chapter 3. Almost Periodic Functions and the Abstract Wave Equation
Abstract
In this chapter, we recall the definition and some basic properties of almost periodic functions with values in a (real) Hilbert space and we establish a necessary and sufficient condition for all solutions of (2.1.7)–(2.1.8) to be almost periodic: \(\mathbb {R} \rightarrow {V}\).
Alain Haraux
Chapter 4. The Wave Equation in a Bounded Domain
Abstract
Let \(\Omega \subset \mathbb {R}^{n}\), \({n} \ge 1\) be a bounded open set. In this section, we study the ordinary wave equation with homogeneous boundary conditions on \(\Gamma = \partial \Omega \).
Alain Haraux
Chapter 5. The Initial-Value Problem For A Mildly Perturbed Wave Equation
Abstract
Let \(\Omega \subset \mathbb {R}^{n}\), \({n} \ge {1}\) be a bounded open set. As a preliminary step towards more complicated situations, in this chapter we study the initial-value problem associated to semi-linear wave equations of the form.
Alain Haraux
Chapter 6. The Initial-Value Problem in Presence of a Strong Dissipation
Abstract
Let \(\Omega \) be a bounded open subset of \(\mathbb {R}^{n}\), \({n} \ge {1}\). In this section, we study the initial-value problem associated to the semi-linear wave equation.
Alain Haraux
Chapter 7. Solutions on and Boundedness of the Energy
Abstract
Let \(\Omega \) be as in Chap. 6 as well as f, g, h. In this chpater we consider the problem on the half-line \(\{{t} \ge 0\}\).
Alain Haraux
Chapter 8. Existence of Forced Oscillations
Abstract
In this chapter, \(\Omega ,{f},{g}\) and h are as in Chaps. 6 and 7. We consider the special case where h is either periodic or (in the last section) almost periodic with respect to t.
Alain Haraux
Chapter 9. Stability of Periodic or Almost Periodic Solutions
Abstract
In this chapter we study the asymptotic behavior of solutions as \({t } \rightarrow + \infty \), mainly in the case where \({h:}\ \mathbb {R} \rightarrow {H}\) is periodic or more generally almost periodic. As already mentioned in Remark 8.4.2, essentially nothing is known in this direction if f is non-linear. Therefore we restrict ourselves to the purely dissipative case. We start with a general result.
Alain Haraux
Chapter 10. The Conservative Case in One Spatial Dimension
Abstract
In this chapter we set \(\Omega = ]0,\ell [\), \(\ell > 0\) and we consider the semilinear problem.
Alain Haraux
Chapter 11. The Conservative Case in Several Spatial Dimensions
Abstract
Let \({n} \ge 1\) and \(\Omega \) any bounded domain in \(\mathbb {R}^{n}\). In this section we consider the semilinear problem.
Alain Haraux
Chapter 12. Recent Evolutions and Perspectives
Abstract
In this chapter, we provide a concise overview of some topics ralated to the main points of this survey that became prominent between 1986 and 2017. Concerning the initial value problem, nothing essentially new happened.
Alain Haraux
Backmatter
Metadata
Title
Nonlinear Vibrations and the Wave Equation
Author
Dr. Alain Haraux
Copyright Year
2018
Electronic ISBN
978-3-319-78515-8
Print ISBN
978-3-319-78514-1
DOI
https://doi.org/10.1007/978-3-319-78515-8

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