  We investigate the long term behavior for a class of competition-diffusion
systems of Lotka-Volterra type for two competing species in the case of low
regularity assumptions on the data. Due to the coupling that we consider the
system cannot be reduced to a single equation yielding uniform estimates with
respect to the inter-specific competition rate parameter. Moreover, in the
particular but meaningful case of initial data with disjoint support and
Dirichlet boundary data which are time-independent, we prove that as the
competition rate goes to infinity the solution converges, along with suitable
sequences, to a spatially segregated state satisfying some variational
inequalities.
