Workshop

ON MODERN MATHEMATICAL

ANALYSIS AND APPLICATIONS

*******

Topic 2:

Abstract Parabolic Evolution Equations

and their applications

Prof. Atsushi Yagi

Graduate School of Engineering, Osaka University, Japan

From 24.08.2009 to 28.08.2009

at the Lecture Room 416 (4-th .oor), House T1,

334 Nguyen Trai str., Hanoi.

This lesson is intended to present the fundamentals of the theory of abstract parabolic

evolution equations and their applications to the various di.usion models in the real world.

Mainly we will follow the semigroups methods which are based on the complex analysis

for the holomorphic functions defined in the complex domain with values in a Banach space.

The analytic semigroups or the evolution operators give us a usful representing formula for

the solutions of abstract parabolic evolution equations from which we can deduce necessary

information about the solutions.

We will focus nonlinear evolution equations, i.e., semilinear equations and quasilinear

equations.

In the first stage, we are concerned with constructing the unique, local or global solutions

to the Cauchy problems for nonlinear abstract evolution equations by utinizing the semigroups

or the evolution operators.

In the secong stage, our attentions are addressed to investigating the dynamical systems

determined from Cauchy systems in an infinite-dimensional space and to analizing the solu-

tions of the Cauchy problems by numerical methods.