### Hiroshi Matano

## Blow-up in Nonlinear Heat Equations

In some classes of nonlinear heat equations, solutions
may develop some kind of singularities in finite time
and may not be continued beyond that time.
Such a phenomenon is called {\it blow-up} and is known
to occur in various types of equations.

There have been many studies on blow-up solutions,
including the pioneering work of H. Fujita in late 1960's,
but it was only in 1980's that people started to understand
the detailed profile of singluarities that occur at the
blow-up time.

In this article I will survey some of the recent works by
various people on the nature of singularities arising in
nonlinear heat equations. It turns out that the dynamical
systems theory can play an important role in those studies,
not always as a rigorous mathematical tool but at least
as a reliable guiding principle. I will focus on the
following three topics:

#### 1. Local structure of singularities

The nature of blow-up can be studied by an asymptotic expansion
of solutions near the blow-up point. In a typical case,
the first term of the asymptotic expansion tells the
blow-up rate, and the second term reveals the local
spatial structure of the singularity. The center manifold
theory gives a deep insight into the problem.

#### 2. Type II blow-up

In equations with some
self-similar structure, one could make a naive guess
on the blow-up rate by simply looking at what
self-similarity the equation has. However, there are
cases in which the actual blow-up rate is faster
(sometimes much faster) than the suspected rate. We will
call such kind of blow-up a `type II' blow-up (some people
call it a `fast' blow-up), whose blow-up rate is generally
very difficult to determine. I will discuss two equations
- a curve shortening equation and a two-dimensional
harmonic map heat flow.

#### 3. Global attractor involving blow-up solutions

When a semilinear diffusion (or heat) equation has a
bounded absorbing set, then its global attractor exists,
and it is the maximal compact invariant set. On the
other hand, if some solutions do not remain bounded,
in particular if some of them blow-up, then such a global
attractor naturally does not exist. In a one-dimensional
diffusion equation, however, there are cases in which one
can extend the notion of global attractor, so as to
incorporate blow-up phenomena into the attractor.

*
Hiroshi Matano*

May 22 1998