Almost periodic solutions of the parabolic-elliptic Keller–Segel system on the whole space

In this paper, we investigate the existence and uniqueness of almost periodic solutions for the parabolic-elliptic Keller–Segel system on the whole space \(\mathbb {R}^n\,\, (n \geqslant 4)\) ${\mathbb{R}}^n(n⩾4)$ . We work in the framework of critical spaces such as on the weak-Lorentz space \(L^{\frac{n}{2},\infty }(\mathbb {R}^n)\) $L^{\frac{n}{2},\infty }({\mathbb{R}}^n)$ . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments.