  In this paper, we characterize a degenerate PDE as the gradient flow in the
space of nonnegative measures endowed with an optimal transport-growth metric.
The PDE of concern, of Hele-Shaw type, was introduced by Perthame et. al. as a
mechanical model for tumor growth and the metric was introduced recently in
several articles as the analogue of the Wasserstein metric for nonnegative
measures. We show existence of solutions using minimizing movements and show
uniqueness of solutions on convex domains by proving the Evolutional
Variational Inequality. Our analysis does not require any regularity assumption
on the initial condition. We also derive a numerical scheme based on the
discretization of the gradient flow and the idea of entropic regularization. We
assess the convergence of the scheme on explicit solutions. In doing this
analysis, we prove several new properties of the optimal transport-growth
metric, which generally have a known counterpart for the Wasserstein metric.
