Cauchy problems and applications

Chin-Yuan Lin

DOI: http://dx.doi.org/10.12775/TMNA.2000.024

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.

Keywords


Cauchy problems; operator semigroups

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