### Cauchy problems and applications

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.

$$

\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

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.