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:
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.
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.
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