Measure valued solutions of sub-linear diffusion equations with a drift term

In this paper we study nonnegative, measure-valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by a strictly increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance. Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass $m_{\rm c}$, which can be explicitly characterized in terms of $\beta$ and of the drift term. If the initial mass is less then $m_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass $m$ of the solutions is greater than the critical one, the stationary solution has a singular part in which the exceeding mass $m- m_{\rm c}$ is accumulated.