We establish the boundedness of solutions of reaction-diffusion systems
with quadratic (in fact slightly superquadratic) reaction terms that
satisfy a natural entropy dissipation property, in any space dimension
. This bound implies
the
-regularity
of the solutions. This result extends the theory which was restricted to the
two-dimensional case. The proof heavily uses De Giorgi’s iteration scheme, which
allows us to obtain local estimates. The arguments rely on duality reasoning in
order to obtain new estimates on the total mass of the system, both in the
norm and in a suitable weak norm. The latter uses
regularization
properties for parabolic equations.
Keywords
reaction-diffusion systems, global regularity, blow-up
methods