Dissipative reaction diffusion systems with quadratic growth

We introduce a class of reaction diffusion systems of which weak solution exists global-in-time with relatively compact orbit in L 1. Reaction term in this class is quasi-positive, dissipative, and up to with quadratic growth rate. If the space dimension is less than or equal to two, the solution is classical and uniformly bounded. Provided with the entropy structure, on the other hand, this weak solution is asymp-totically spatially homogeneous.