Quasilinear evolution equations with non-densely defined operators

Abstract

Two problems for the abstract quasilinear evolution equation of "hyperbolic" type in a Banach space $$ u'(t) = A(t,u(t))u(t) \,\,\,\rm{for} \,\,\,t \geq 0, \,\,\, \rm{and} \,\,\, u(0)=u_0 $$ are studied without assuming that the domain of $A(t,w)$ is dense in the whole space. One is the fundamental problem of existence and uniqueness of classical solutions, and the other is the problem of extension or blow up of classical solutions.