Parabolic-elliptic and indirect-direct simplifications in chemotaxis systems driven by indirect signalling

Abstract

Singular limits for the following indirect signalling chemotaxis system \begin{align*} \left\{ \begin{array}{lllllll} \partial_t n = \Delta n - \nabla \cdot (n \nabla c ) & \text{in } \Omega\times(0,\infty) , \varepsilon \partial_t c = \Delta c - c + w & \text{in } \Omega\times(0,\infty), \varepsilon \partial_t w = \tau \Delta w - w + n & \text{in } \Omega\times (0,\infty), \partial_\nu n = \partial_\nu c = \partial_\nu w = 0, &\text{on } \partial\Omega\times (0,\infty) %(n,c,w)_{t=0} = (n_0,c_0,w_0) & \text{on } \Omega, \end{array} \right. \end{align*} are investigated. More precisely, we study parabolic-elliptic simplification, or PES, $\varepsilon\to 0^+$ with fixed $\tau>0$ up to the critical dimension $N=4$, and indirect-direct simplification, or IDS, $(\varepsilon,\tau)\to (0^+,0^+)$ up to the critical dimension $N=2$. These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the $L^p$-energy function is devised, which allows us to obtain suitable uniform bounds for the singular limits. Moreover, in both scenarios, we also present the convergence rates, where the effect of the initial layer and the convergence to the critical manifold are also revealed.