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# Global solutions for
some reaction-diffusion systems: L^{∞},L^{p},
L^{1} and L^{2}-strategies

Abstract:
Lots of reaction-diffusion systems arising in applications come with the two natural following properties:

- Positivity of the solutions is preserved for all time
- The total mass of the components is controlled for all time.

The fact that the
total mass of the components does not blow up in finite time suggests
that solutions should exist for all time (solutions are actually
bounded in L^{1}, uniformly in time). But, it turns out that
the answer is not so simple. An L^{p}-strategy provides global
existence of classical solutions for a subfamily of these systems.
But, to handle more of them, it is necessary to give up looking for
bounded classical solutions and rather consider *weak solutions*
which may be allowed to blow up in L^{∞} at some
time(s), and nevertheless continue to exist. For instance, global
weak solutions exist as soon as the nonlinearities are bounded in
L^{1}.

A curious L^{2}-estimate is a priori valid
for all these systems. This estimate allows to prove global existence
when nonlinearities are at most quadratic as it is the case for many
chemical and biological systems. Actually, an a priori
L^{2}-compactness even holds: it turns out to be an adequate
tool to study the limit of chemical systems where some rate constants
tend to infinity.

We will give a survey on these questions and
describe some recent results together with open
problems.