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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Unbounded Linear Operators and Evolution Equations

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

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

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

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

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

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

Let $$\Omega$$ be as in Chap. 6 as well as fgh. In this chpater we consider the problem on the half-line $$\{{t} \ge 0\}$$.
Alain Haraux

### Chapter 8. Existence of Forced Oscillations

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

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

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

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

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

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