Strong competition versus fractional diffusion: the case of Lotka-Volterra interaction

Abstract

We consider a system of differential equations with nonlinear Steklov boundary conditions, related to the fractional problem $$(-\Delta {)}^s{u}_i=f_i(x,u_i)-\beta u_i^p\sum _{j\ne i}{a}_{ij}{u}_j^p,$$ where $i=i,\dots ,k$, $s\in (0,1)$, $p>0$, $a_{ij}>0$ and $\beta >0$. When $k=2$ we develop a quasi-optimal regularity theory in $C^{0,\alpha }$, uniformly w.r.t. $\beta$, for every $\alpha <{\alpha }_{\mathrm{o}pt}=min(1,2s)$; moreover we show that the traces of the limiting profiles as $\beta \to +\infty$ are Lipschitz continuous and segregated. Such results are extended to the case of $k\ge 3$ densities, with some restrictions on $s$, $p$ and $a_{ij}$.

References in corpus (1)

• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian

Cited by in corpus (1)

• On the uniqueness of solutions of an nonlocal elliptic system