We discuss global existence and asymptotic behaviour of a price formation free boundary model introduced by Lasry and Lions in 2007. Our results are based on a construction which transforms the problem into the heat equation with specially prepared initial datum. The key point is that the free boundary present in the original problem becomes the zero level set of this solution. Using the properties of the heat operator we can show global existence, regularity and asymptotic results of the free boundary. 2011 Académie des sciences.
We acknowledge supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST), by the Leverhulme Trust via Kinetic and mean field partial differential models for socio-economic processes (PI Peter Markowich) and by the Royal Society through the Wolfson Research Merit Award of Peter Markowich. Luis Caffarelli acknowledges support from the Division of Mathematical Sciences of the NSF.