Existence and asymptotic stability in a fractional chemotaxis system with competitive kinetics

ABSTRACT

This paper studies a fully parabolic chemotaxis system with competitive kinetics and fractional diffusion of order ${\alpha }_1,{\alpha }_2\in (0,2)$ $\{\begin{array}{ll}{u}_t=-{\mathrm{d}}_1{\mathrm{\Lambda }}^{{\alpha }_1}u-{\chi }_1\mathrm{\nabla }\cdot \left(u\mathrm{\nabla }w\right)+{\mu }_{1}u(1-u-a_{1}v),& x\in {\mathbb{T}}^2,t>0,\\ v_t=-{\mathrm{d}}_2{\mathrm{\Lambda }}^{{\alpha }_2}v-{\chi }_2\mathrm{\nabla }\cdot \left(v\mathrm{\nabla }w\right)+{\mu }_{2}v(1-v-a_{2}u),& x\in {\mathbb{T}}^2,t>0,\\ w_t={\mathrm{d}}_3\mathrm{\Delta }w-{\gamma }_{1}w+{\gamma }_{2}u+{\gamma }_{3}v,& x\in {\mathbb{T}}^2,\end{array}$ on two dimensional periodic torus ${\mathbb{T}}^2$. It is proved that (1(1) $\{\begin{array}{ll}{u}_t=-{\mathrm{d}}_1{\mathrm{\Lambda }}^{{\alpha }_1}u-{\chi }_1\mathrm{\nabla }\cdot \left(u\mathrm{\nabla }w\right)+{\mu }_{1}u(1-u-a_{1}v),& x\in {\mathbb{T}}^2,t>0,\\ v_t=-{\mathrm{d}}_2{\mathrm{\Lambda }}^{{\alpha }_2}v-{\chi }_2\mathrm{\nabla }\cdot \left(v\mathrm{\nabla }w\right)+{\mu }_{2}v(1-v-a_{2}u),& x\in {\mathbb{T}}^2,t>0,\\ w_t={\mathrm{d}}_3\mathrm{\Delta }w-{\gamma }_{1}w+{\gamma }_{2}u+{\gamma }_{3}v,& x\in {\mathbb{T}}^2,t>0,\\ u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),w(x,0)=w_0\left(x\right),& x\in {\mathbb{T}}^2,\end{array}$(1) ) has a unique global bounded solution for all appropriately regular nonnegative initial data $u_0,v_0$ and $w_0$. Moreover, if ${\mu }_1$ and ${\mu }_2$ (${\mu }_1$ or ${\mu }_2$) are large enough, we can reach the following conclusions by constructing appropriate energy functional:

(i) $0<a_1,a_2<1$ $(u,v,w)(\cdot ,t)\to (\frac{1-a_1}{1-a_1{a}_2},\frac{1-a_2}{1-a_1{a}_2},\frac{{\gamma }_2(1-a_1)+{\gamma }_2(1-a_2)}{{\gamma }_1(1-a_1{a}_2)})$ uniformly in ${\mathbb{T}}^2$ as $t\to \mathrm{\infty }$.

(ii) $a_1\ge 1,0<a_2<1$ $(u,v,w)(\cdot ,t)\to (0,1,\frac{{\gamma }_3}{{\gamma }_1})$ uniformly in ${\mathbb{T}}^2$ as $t\to \mathrm{\infty }$.

(iii) $0<a_1<1,a_2\ge 1$ $(u,v,w)(\cdot ,t)\to (1,0,\frac{{\gamma }_2}{{\gamma }_1})$ uniformly in ${\mathbb{T}}^2$ as $t\to \mathrm{\infty }$.

KEYWORDS:

• Chemotaxis system
• fractional diffusion
• competitive kinetics
• global classical solution
• asymptotic stability

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35B40
• 35K55
• 35Q92
• 92C17

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is partially supported by National Natural Science Foundation of China [grant number 11771380] and Natural Science Foundation of Jiangsu Province [grant number BK20191436].