Electronic Research Archive

2021, Issue 5: 3261-3279. doi: 10.3934/era.2021037

Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop

• Received: 01 February 2021 Revised: 01 April 2021 Published: 26 May 2021
• Primary: 35A01, 35K55; Secondary: 92C17

• In this work, the fully parabolic chemotaxis-competition system with loop

$\begin{eqnarray*} \left\{ \begin{array}{llll} &\partial_{t} u_{1} = d_1\Delta u_{1}-\nabla\cdot(u_{1}\chi_{11}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{1}\chi_{12}(v_{2})\nabla v_{2}) +\mu_{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ &\partial_{t} u_{2} = d_2\Delta u_{2}-\nabla\cdot(u_{2}\chi_{21}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{2}\chi_{22}(v_{2})\nabla v_{2}) +\mu_{2}u_{2}(1-u_{2}-a_{2}u_{1}), \\ &\partial_t v_1 = d_3\Delta v_{1}-\lambda_{1} v_{1}+h_1(u_{1}, u_{2}), \\ &\partial_t v_2 = d_4\Delta v_{2}-\lambda_{2} v_{2}+h_2(u_{1}, u_{2}) \\ \end{array} \right. \end{eqnarray*}$

is considered under the homogeneous Neumann boundary condition, where $x\in\Omega, t>0$, $\Omega\subset \mathbb{R}^{n} (n\leq 3)$ is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters $\mu_1, \mu_2$ are sufficiently large, then the system possesses a unique and global classical solution for $n\leq 3$. Specifically, when $n = 2$, the global boundedness can be attained without any constraints on $\mu_1, \mu_2$.

Citation: Chun Huang. Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop[J]. Electronic Research Archive, 2021, 29(5): 3261-3279. doi: 10.3934/era.2021037

Related Papers:

• In this work, the fully parabolic chemotaxis-competition system with loop

$\begin{eqnarray*} \left\{ \begin{array}{llll} &\partial_{t} u_{1} = d_1\Delta u_{1}-\nabla\cdot(u_{1}\chi_{11}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{1}\chi_{12}(v_{2})\nabla v_{2}) +\mu_{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ &\partial_{t} u_{2} = d_2\Delta u_{2}-\nabla\cdot(u_{2}\chi_{21}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{2}\chi_{22}(v_{2})\nabla v_{2}) +\mu_{2}u_{2}(1-u_{2}-a_{2}u_{1}), \\ &\partial_t v_1 = d_3\Delta v_{1}-\lambda_{1} v_{1}+h_1(u_{1}, u_{2}), \\ &\partial_t v_2 = d_4\Delta v_{2}-\lambda_{2} v_{2}+h_2(u_{1}, u_{2}) \\ \end{array} \right. \end{eqnarray*}$

is considered under the homogeneous Neumann boundary condition, where $x\in\Omega, t>0$, $\Omega\subset \mathbb{R}^{n} (n\leq 3)$ is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters $\mu_1, \mu_2$ are sufficiently large, then the system possesses a unique and global classical solution for $n\leq 3$. Specifically, when $n = 2$, the global boundedness can be attained without any constraints on $\mu_1, \mu_2$.

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