On Global Solutions and Blow-Up for a Short-Ranged Chemical Signaling Loop

Abstract

This paper aims at providing a substantial step toward the global boundedness and blow-up of solutions to a two-species and two-stimuli chemotaxis model, in which the process of the species results in a short-ranged chemical signaling loop. More precisely, we consider the following Neumann initial-boundary value problem

$$\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta u-\chi _1\nabla \cdot (u\nabla v), &{} \quad x\in \varOmega , &{}t>0,\\ 0=\Delta v-\mu _2+w, &{} \quad x\in \varOmega ,&{} t>0,\\ w_t=\Delta w-\chi _2\nabla \cdot ( w\nabla z)-\chi _3\nabla \cdot ( w\nabla v), &{} \quad x\in \varOmega , &{}t>0,\\ 0=\Delta z-\mu _1+u, &{} \quad x\in \varOmega , &{}t>0\\ \end{array} \right. \end{aligned}$$

in the unit disk $\varOmega :=B_1(0)\subset \mathbb {R}^2$ with chemotactic sensitivities $\chi _1>0,\chi _2>0$ and $\chi _3\ge 0$ and radially symmetric nonnegative initial data $u_0$ and $w_0$, where $\mu _1 and \mu _2$ are given, respectively, by $\mu _1:=\frac{m_1}{|\varOmega |}$ and $\mu _2:=\frac{m_2}{|\varOmega |}$ with $m_1=\int _{\varOmega }u_0$, $m_2=\int _{\varOmega }w_0$.

Explicit conditions on $\chi _i, \mu _i$, $u_0$ and $w_0$ are given for the simultaneous global boundedness and simultaneous finite-time blow-up of classical solutions. Specifically, when the effect of $\chi _3>0$ is strong enough in the sense that the dynamical properties of the above system behave like one single-species Keller–Segel chemotaxis system, it is shown that if only the total mass $m_2<\frac{4\pi }{\chi _1}$ and $m_2<\frac{4\pi }{\chi _3}$, the solutions are globally bounded, while blow-up may occur provided that $m_2>\frac{4\pi }{\chi _1}$ and $m_2>\frac{8\pi }{\chi _3}$. Moreover, in view of the chemotactic signaling loop, one can find a critical mass phenomenon:

• if $2m_1+\frac{m_2\chi _3}{\chi _2}<\frac{8\pi }{\chi _2}$, the problem possesses only globally bounded solutions, whereas

• if $\frac{8\pi m_1}{\chi _1}+\frac{8\pi m_2}{\chi _2}<2m_1m_2+\frac{m^2_2\chi _3}{\chi _2}$ and $\int _{\varOmega }u_0|x|^2$ and $\int _{\varOmega }w_0|x|^2$ are sufficiently small, the corresponding solution blows up in finite time in the sense that

  $$\begin{aligned} \limsup _{t\nearrow T}\left( ||u(\cdot ,t)||_{L^{\infty }(\varOmega )}+||w(\cdot ,t)||_{L^{\infty }(\varOmega )}\right) \rightarrow \infty \end{aligned}$$

with some finite time $T>0$. The simultaneous blow-up phenomenon for two species is also given in this paper.

This, in particular, shows, when $\chi _3=0$, that smallness of mass of each species implies global solvability, whereas largeness of masses induces blow-up to occur.