2 results
The fully parabolic multi-species chemotaxis system in
$\mathbb{R}^{2}$
- Part of
-
- Journal:
- European Journal of Applied Mathematics / Volume 35 / Issue 5 / October 2024
- Published online by Cambridge University Press:
- 19 January 2024, pp. 675-706
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
This article is devoted to the analysis of the parabolic–parabolic chemotaxis system with multi-components over
$\mathbb{R}^2$. The optimal small initial condition on the global existence of solutions for multi-species chemotaxis model in the fully parabolic situation had not been attained as far as the author knows. In this paper, we prove that under the sub-critical mass condition, any solutions to conflict-free system exist globally. Moreover, the global existence of solutions to system with strong self-repelling effect has been discussed even for large initial data. The proof is based on the modified free energy functional and the Moser–Trudinger inequality for system.
On the global existence of solutions to chemotaxis system for two populations in dimension two
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 6 / December 2023
- Published online by Cambridge University Press:
- 09 January 2023, pp. 2106-2128
- Print publication:
- December 2023
-
- Article
-
- You have access
- HTML
- Export citation
-
We consider the global existence for the following fully parabolic chemotaxis system with two populations
\[\left\{ \begin{array}{@{}ll} \partial_tu_i=\kappa_i\Delta u_i-\chi_i\nabla\cdot(u_i\nabla v),\quad i\in\{1,2\}, & x\in\Omega,\ t>0, \\ v_t=\Delta v-v+u_1+u_2, & x\in\Omega,\ t>0,\\ u_i(x,t=0)=u_{i0}(x),\quad v(x,t=0)=v_0(x), & x\in\Omega, \end{array} \right. \]where $\Omega =\mathbb {R}^2$
or $\Omega =B_R(0)\subset \mathbb {R}^2$
supplemented with homogeneous Neumann boundary conditions, $\kappa _i,\chi _i>0,$
$i=1,2$
. The global existence remains open for the fully parabolic case as far as the author knows, while the existence of global solution was known for the parabolic-elliptic reduction with the second equation replaced by $0=\Delta v-v+u_1+u_2$
or $0=\Delta v+u_1+u_2$
. In this paper, we prove that there exists a global solution if the initial masses satisfy the certain sub-criticality condition. The proof is based on a version of the Moser–Trudinger type inequality for system in two dimensions.
