Quasi-linear evolution equations in non-reflexive banach spaces, I

Abstract

IN HIS PAPER ‘Quasi-linear equations of evolution, with applications to partial differential equations’, Kato [7] has presented a unified treatment of the Cauchy problem for various quasilinear partial differential equations that appear in mathematical physics, based on the theory of abstract equations of evolutions. He proved there two very general theorems on abstract equations: one on the existence and uniqueness, and the other on the continuous dependence on data of the solution. Recently, Dorroh and Graff [S] developed a new approach to linear evolution systems which is based on two new concepts: weak evolution systems and approximate solutions. Partially based on these results is a recent general existence and uniqueness theorem for quasi-linear evolution equations proved by Graff [6]. This paper presents an entirely different approach to quasi-linear evolution equations which is not based directly on the C,-semigroup theory. First of all this approach is based on the theory of contractor directions. However, a straightforward application of the method of contractor directions is in general of no use. Therefore, a considerable further development of the theory involved was necessary. Along this line two new ideas combined contributed to our new approach. First let us mention the introduction of the concept of the Lipschitz commutator property of the family {A(t, x)} of linear continuous mappings of Y c X into X, i.e., that