Studied Systems and General Results

Abstract

This chapter is devoted to general tools and basic results, on the existence, uniqueness and regularity of the solutions of linear time-dependent systems, which will be used frequently in later chapters.

For the description of various function spaces including Sobolev spaces and different notations, the reader is referred to Chapter 3 for details.

Our objective in this chapter is to recall the general framework and the basic results of existence of solutions for linear evolutive equations of the first order in time of the type

$$ \frac{\partial u}{\partial t} + A(t)u = f(t) $$

.

We consider some operator A (time dependent) and a right-hand side f which is allowed to depend on time. First we give some hypotheses and properties, and second we indicate in what sense such an equation is understood and give the existence, the uniqueness and the regularity of solutions.