Minimizing movements for quasilinear Keller--Segel systems with nonlinear mobility in weighted Wasserstein metrics

Abstract

We prove the global existence of weak solutions to quasilinear Keller--Segel systems with nonlinear mobility by minimizing movements (JKO scheme) in the product space of the weighted Wasserstein space and $L^2$ space. In particular, we newly show the global existence of weak solutions to the Keller--Segel system with the degenerate diffusion and the sub-linear sensitivity in the critical case. The advantage of our approach is that we can connect the global existence of weak solutions to the Keller--Segel systems with the boundedness from below of a suitable functional. While minimizing movements for Keller--Segel systems with linear mobility are adapted in the product space of the Wasserstein space and $L^2$ space, due to the nonlinearity of mobility, we need to use the weighted Wasserstein space instead of the Wasserstein space. Moreover, since the mobility function is not Lipschitz, we first find solutions to the Keller--Segel systems whose mobility is approximated by a Lipschitz function, and then we establish additional uniform estimates and convergences to derive solutions to the Keller--Segel systems.