  The aim of this paper is to provide the analysis result for the partial
differential equations arising from the rigorous derivation of the degenerate
parabolic-elliptic Keller-Segel system from a moderately interacting stochastic
particle system. The rigorous derivation is divided into two articles. In this
paper, we establish the solution theory of the degenerate parabolic-elliptic
Keller-Segel problem and its non-local version, which will be used in the
second paper for the discussion of the mean-field limit. A parabolic
regularized system is introduced to bridge the stochastic particle model and
the degenerate Keller-Segel system. We derive the existence of the solution to
this regularized system by constructing approximate solutions, giving uniform
estimates and taking the limits, where a crucial step is to obtain the L infty
Bernstein type estimate for the gradient of the approximate solution. Based on
this, we obtain the well-posedness of the corresponding non-local equation
through perturbation method. Finally, the weak solution of the degenerate
Keller-Segel system is obtained by using a nonlinear version of Aubin-Lions
lemma.
