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# Yannick Sire

## Qualitative properties of reaction-diffusion equations

## курс отменён

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.