N-barrier maximum principle for degenerate elliptic systems and its   application
  In this paper, we prove the N-barrier maximum principle, which extends the result in [5] from linear diffusion equations to nonlinear diffusion equations, for a wide class of degenerate elliptic systems of porous medium type. The N-barrier maximum principle provides a priori upper and lower bounds of the solutions to the above-mentioned degenerate nonlinear diffusion equations including the Shigesada-Kawasaki-Teramoto model as a special case. As an application of the N-barrier maximum principle to a coexistence problem in ecology, we show the nonexistence of waves in a three-species degenerate elliptic systems. 
Li-Chang Hung, Hsiao-Feng Liu, Chiun-Chuan Chen
math.AP