Today post is not a new ArXiv article but instead referred from such an article. While reading the Introduction of this (today) ArXiv paper of Dung Le, I came to another old ArXiv paper of him which I’m going to write about (very shortly) now.
The author studied strongly coupled parabolic system of the form
with initial data and boundary condition, in which where is a bounded domain in .
The diffusion matrix and nonlinearity satisfy some natural conditions. For example, one can take as a linear matrix satisfying the elliptic condition and to be polynomial.
Assuming that and initial data , it is well-known that the system has a unique local classical solution
Moreover, the criterion for blow-up is: if then
In other words, if we can prove that
then , that is the global solution exists globally.
Note that since , thanks to the Morrey embedding the condition (*) implies that the solution is bounded in . This fact is usually very hard to prove in general system due to the lack of comparison principle.
In this paper, the author succeeded in providing a weaker criterion for blow-up, or equivalently global existence of solution, that is the solutions needs only to be bounded in -norm. To be more precise, the BMO-norm (Bounded Mean Oscillation) is defined as follow
where and is the mean value of over .
where is a continuous function on , and for any and , there exists such that
As a corrollary, if
then the classical solution exists globally.