Stability of invasion traveling waves for a competition system with nonlocal dispersals

In this paper, we mainly investigate the stability of invasion traveling waves for a competition system with nonlocal dispersals. We prove that the invasion traveling waves are exponentially stable as perturbation in some exponentially weighted spaces, when the difference between initial data and traveling waves decays exponentially as x -> -infinity, but in other locations, the initial data can be very large. The adopted method is to use the weighted energy and the squeezing technique with some new flavors to handle the nonlocal dispersals.