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Yannick Sire

Qualitative properties of reaction-diffusion equations

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Several physical phenomena are governed by nonlinear partial differential equations. Contrary to ordinary differential equations, the problems of existence and regularity of the solutions cannot be done in a unified way and one has to study each particular example. In this course, we want to focus on partial differential equations arising in many areas like combustion theory, population dynamics and biology. At the simplest level of modelization, these phenomena are described by reaction-diffusion equations.

The theory of this type of equation is far away from being fully understood but one can give however several results providing in some cases existence of solutions and their qualitative behaviour. An interesting property of these equations are that they carry a nice geometric insight.

1. Reaction-diffusion equations. Travelling wave solutions. Existence of travelling waves in the 1D framework.

2. Theory of uniformly elliptic equations: existence, regularity, Schauder theory, Agmon-Douglis-Nirenberg theory, Harnack estimates, boundary Harnack principle.

3. Elements of the theory of parabolic equations (semi-group theory). Sectorial operators.

4. Existence of multi-dimensional travelling waves. Proof of the Berestycki-Larrouturou-Lions theorem. Stability properties.

5. Free boundary problem: non degeneracy near the free boundary, optimal regularity, free boundary condition, Hausdorff estimates (Berestycki-Caffarelli-Nirenberg theory). Application to the flame propagation model in the limit of high activation energies.

If time permits, we might develop several topics related to research: homogenization, travelling waves in periodic media, integral diffusion, theory of free boundaries.