ON EXISTENCE AND UNIQUENESS IN A FREE BOUNDARY PROBLEM FROM COMBUSTION

ABSTRACT

We study a free boundary problem for the heat equation describing the propagation of laminar flames under certain geometric assumptions on the initial data. The problem arises as the limit of a singular perturbation problem, and generally no uniqueness of limit solutions can be expected. However, if the initial data is starshaped, we show that the limit solution is unique and coincides with the minimal classical supersolution. Under certain convexity assumption on the data, we prove first that the limit solution is a classical solution of the free boundary problem for a short time interval, and then that the solution, in fact, stays classical as long as it does not vanish identically.

Keywords:

• 1991 Mathematics Subject Classification.
• Primary 35R35
• 35K05
• Parabolic free boundary problem
• Singular perturbation
• Limit solution
• Classical solution
• Uniqueness
• Existence

ACKNOWLEDGMENT

The author thanks Luis Caffarelli for a number of fruitful discussions. Without his help and support this paper would never exist.