# American Institute of Mathematical Sciences

March  2016, 21(2): 417-436. doi: 10.3934/dcdsb.2016.21.417

## Stefan problem, traveling fronts, and epidemic spread

 1 Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen

Received  November 2014 Revised  April 2015 Published  November 2015

The scalar reaction diffusion equation with a nonlinearity of logistic type has a minimal speed $c_0$ for standard traveling fronts. It is shown that also for speeds $0 < c < c_0$ there are traveling fronts but these are solutions to free boundary value (Stefan) problems. Furthermore, these speeds depend in a monotone way on the Stefan coefficient which links the loss of matter at the free boundary to the displacement per time. The results are extended to correlated random walks, Cattaneo systems and, in particular, to models for epidemic spread. In the epidemic problems a dichotomy phenomenon shows up: For small values of the Stefan coefficient there are no fronts indicating that for such values and certain data the free boundary stays bounded.
Citation: Karl P. Hadeler. Stefan problem, traveling fronts, and epidemic spread. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 417-436. doi: 10.3934/dcdsb.2016.21.417
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