Cauchy problems and applications

Abstract

Of concern is the Cauchy problem $$ \frac{du}{dt} \in Au,\quad u(0) = u_{0},\quad t > 0, $$ where $ u : [0, \infty) \to X$, $X $ is a real Banach space, and $ A : D(A) \subset X \to X $ is nonlinear and multi-valued. It is showed by the method of lines, combined with the Crandall-Liggett theorem that this problem has a limit solution, and that the limit solution is a unique strong one if $ A $ is what is called embeddedly quasi-demi-closed. In the case of linear, single-valued $ A $, further results are given. An application to nonlinear partial differential equations in non-reflexive $ X $ is given.