Seminar 29.2.2024: Philippe Souplet

[quittner]

Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations

Štvrtok 29.2.2024 o 14:00, poslucháreň M-223

Philippe Souplet (Université Sorbonne Paris Nord):
Convergence, concentration and critical mass phenomena
for a model of cell migration with signal production on the boundary

Abstract:
We consider a model of cell migration with signal production on the boundary.
It consists in a diffusion equation with nonlinear nonlocal advection,
complemented by a no-flux condition ensuring mass conservation.

For nonlinearities with polynomial growth, we first develop a local
existence-uniqueness theory in optimal $L^p$ spaces. With help of this tool,
we next obtain the following results on the global behavior of solution:
- For small initial data, we have exponential convergence towards a constant.
- If, and only if, the growth of the nonlinearity is at least quadratic,
we have concentration, i.e.~finite time blowup, for large initial data.
- In the critical case of a quadratic nonlinearity, we observe a critical
mass phenomenon {\it in any space dimension} (denoting by $M$ the mass
of the $L^1$ initial data):
-- for $M\le 1$, the solution is global and bounded;
-- for $M>1$, there exist initial data leading to finite time concentration.
This critical mass phenomenon is reminiscent of the well-known situation
for the 2D Keller-Segel system. The global existence proof is delicate,
based on a control of the solution by means of an entropy functional,
via an $\varepsilon$-regularity type result.
- Finally we give some partial results on the localization and final profile
of the boundary concentration and on the blowup rate.

This is a joint work with Nicolas Meunier (Universit\'e d'Evry-Val d'Essonne).