The F-KPP equation in the half-plane

par Julien Berestycki

mardi 5 déc. 2023, 09:50 → 10:50 Europe/Paris

Salle Pellos (1R2-207) (IMT)

It has been shown by H. Berestycki and G. Cole (2022) that  the F-KPP equation ${\partial }_{t}u=\frac{1}{2}\mathrm{\Delta }u+u(1-u)$ in the half-plane with Dirichlet boundary conditions admits travelling wave solutions for all speed $c\ge c^{\ast }=\sqrt{2}$.
We show that the minimal speed traveling wave $\mathrm{\Phi }$ is in fact unique (up to shift) and give a probabilistic representation as the Laplace transform of a certain martingale limit associated to the branching Brownian motion with absorption. This representation allows us to study the asymptotic behaviour of $\mathrm{\Phi }$ away from the boundary of the domain, proving that
$$\underset{y\to \mathrm{\infty }}{lim}\mathrm{\Phi }(x+\frac{1}{\sqrt{2}}\log y,y)=w(x)$$
where $w$ is the usual one-dimensional critical travelling wave.
We are able to extend our result to the case of the half-space ${\mathbb{H}}^d=\{x\in R^d:x_1\ge 0\}$. Finally, if time allows, I will also mention some results regarding the convergence towards the critical travelling wave.
This is based on joint work with Cole Graham, Yujin Kim and Bastien Mallein.