Traveling waves for the Keller-Segel-FKPP equation with strong chemotaxis
We show that there exist traveling wave solutions of the Keller-Segel-FKPP equation, which models a diffusing and logistically growing population subject to chemotaxis. In contrast to previous results, our result is in the strong aggregation regime; that is, we make no smallness assumption on the parameters. The lack of a smallness condition makes $L^\infty$-estimates difficult to obtain as the comparison principle no longer gives them ``for free." Instead, our proof is based on suitable energy estimates in a carefully tailored uniformly local $L^p$-space. Interestingly, our uniformly local space involves a scaling parameter, the choice of which is a crux of the argument. Numerical experiments exploring the stability, qualitative properties, and speeds of these waves are presented as well.