The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion

We prove the time-global existence of solutions of the degenerate Keller-Segel system in higher dimensions, under the assumption that the mass of the first component is below a certain critical value. What we deal with is the full parabolic-parabolic system rather than the simplified parabolic-elliptic system. Our approach is to formulate the problem as a gradient flow on the Wasserstein space. We first consider a time-discretized problem, in which the values of the solution are determined iteratively by solving a certain minimizing problem at each time step. Here we use a new minimizing scheme at each time level, which gives the time-discretized solutions favorable regularity properties. As a consequence, it becomes relatively easy to prove that the time-discretized solutions converge to a weak solution of the original system as the time step size tends to zero.