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

Abstract

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 an 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 $m_{\mathrm{c}}$, which can be explicitely characterized in terms of $\beta$ and of the drift term. If the initial mass is less then $m_{\mathrm{c}}$, 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 steady state has a singular part in which the exceeding mass $m-m_{\mathrm{c}}$ is accumulated.

30 pages

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