Quasilinear Evolution Equations

Abstract

This chapter is devoted to the study of semilinear evolution equations of the form

$$u'\left( t \right) = A\left( t \right)u\left( t \right) + f\left( {t,u\left( t \right)} \right) $$

and quasilinear evolution equations of the form

$$u'\left( t \right) = A\left( {t,u\left( t \right)} \right)u\left( t \right) + f\left( {t,u\left( t \right)} \right) $$

in scales of Banach spaces, where A(t) resp., A(t, v) are linear operators. Although semilinear evolution equations are special cases of quasilinear ones, we will start this chapter with a discussion of semilinear evolution equations in scales of Banach spaces in section 3.1. We do this for two reasons. On the one hand, the proofs are much easier and clearer than in the case of quasilinear evolution equations; on the other hand, we also get more precise results in this special case. Then, in section 3.2 we collect some commutator estimates that we will need in the sequel. In 3.3 we give a general uniqueness and existence result for quasilinear evolution equations provided that the Cauchy problem for related time-dependent linear evolutions equations is well-posed. These assumptions will always be satisfied in the situations described in chapter 2. Finally, in 3.4 we will prove a regularity result for these equations in scales of Hilbert spaces.