ABSTRACT CAUCHY PROBLEMS FOR QUASI-LINEAR EVOLUTION EQUATIONS WITH NON-DENSELY DEFINED OPERATORS

Abstract

In this paper we study the abstract Cauchy problem for quasi-linear evolution equation $u'(t) = A(u(t)) u(t)$, where $\{ A(w); w \in W \}$ is a family of closed linear operators in a real Banach space $X$ such that $D(A(w)) = Y$ for $w \in W$, and $W$ is an open subset of another Banach space $Y$ which is continuously embedded in $X$. The purpose of this paper is not only to establish a ‘global’ well-posedness theorem without assuming that $Y$ is dense in $X$ but also to propose a new type of dissipativity condition which is closely related with the continuous dependence of solutions on initial data.