Nonlinear stability of forced traveling waves for a Lotka-Volterra cooperative model under climate change

This paper is concerned with the nonlinear stability of forced traveling waves for a Lotka-Volterra cooperative model under climate change. Firstly, by applying the L-2-weighted energy estimate method, the comparison principle, and the squeezing technique, we investigate that all forced traveling waves with the speed c > (c) over bar are exponentially stable in the form of e(-nu t) for some nu>0. Secondly, in order to improve the former results, we take another weight function and construct the related different weighted Sobolev space. Instead of the L-2-weighted energy estimate, we first establish a L-1-weighted energy estimate in the weighted Sobolev space. Then, by using this crucial L-1-estimate, we further obtain the desired L-2-energy estimate. Finally, we obtain that all forced traveling waves with the speed c>c over bar $$ c>\overline{c} $$ are exponentially stable in the form of (e-mu 3t) for some mu>0, where (c) over bar < (c) over bar.