Dissipative Reaction Diffusion Systems with Quadratic Growth

We introduce a class of reaction diffusion systems of which a weak solution exists globally in time with relatively compact orbit in L¹. The reaction term in this class is quasipositive, 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. If we are provided with the entropy structure, on the other hand, this weak solution is asymptotically spatially homogeneous.