Uniform Hölder estimate for singularly perturbed parabolic systems of Bose–Einstein condensates and competing species

We prove the uniform Hölder continuity of solutions for two classes of singularly perturbed parabolic systems. These systems arise in Bose–Einstein condensates and in competing models in population dynamics. The proof relies upon the blow up technique and the monotonicity formulas by Almgren and Alt, Caffarelli, and Friedman.