We discuss the Cauchy problem for the following parabolic attraction–repulsion chemotaxis system: $\left\{\begin{array}{cc}{\partial }_{t}u=\Delta u-\nabla \cdot ({\beta }_{1}u\nabla v_1)+\nabla \cdot ({\beta }_{2}u\nabla v_2),\hfill & t>0,x\in {\mathbb{R}}^2,\hfill \\ {\partial }_t{v}_j=\Delta v_j-{\lambda }_j{v}_j+u,\hfill & t>0,x\in {\mathbb{R}}^2(j=1,2),\hfill \\ u(0,x)=u_0\left(x\right),v_j(0,x)=v_{j0}\left(x\right),\hfill & x\in {\mathbb{R}}^2(j=1,2)\hfill \end{array}\right.$with constants ${\beta }_j$, ${\lambda }_j>0$ $(j=1,2)$. In this paper we prove that the nonnegative solutions exist globally in time under the assumption $({\beta }_1-{\beta }_2){\int }_{{\mathbb{R}}^2}{u}_{0}dx<8\pi$ in the attractive dominant case ${\beta }_1>{\beta }_2$.

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抛物线-排斥化学趋化系统的解的全球存在 ${\mathbb{电阻}}^2$: 有吸引力的优势案例

我们讨论以下抛物线吸引-排斥趋化系统的柯西问题： $\left\{\begin{array}{cc}{\partial }_{吨}你=\Delta 你-\nabla \cdot ({\beta }_{\mathrm{1个}}你\nabla v_{\mathrm{1个}})+\nabla \cdot ({\beta }_2你\nabla v_2),\hfill & 吨>0,X\in {\mathbb{电阻}}^2,\hfill \\ {\partial }_{吨}{v}_j=\Delta v_j-{\lambda }_j{v}_j+你,\hfill & 吨>0,X\in {\mathbb{电阻}}^2(j=\mathrm{1个},2),\hfill \\ 你(0,X)={你}_0\left(X\right),v_j(0,X)=v_{j0}\left(X\right),\hfill & X\in {\mathbb{电阻}}^2(j=\mathrm{1个},2)\hfill \end{array}\right.$有常数 ${\beta }_j$, ${\lambda }_j>0$ $(j=\mathrm{1个},2)$. 在本文中，我们证明了在假设条件下，非负解在时间上是全局存在的$({\beta }_{\mathrm{1个}}-{\beta }_2){\int }_{{\mathbb{电阻}}^2}{你}_{0}dX<8\pi$ 在有吸引力的优势情况下 ${\beta }_{\mathrm{1个}}>{\beta }_2$.