一类二维Keller-Segel系统整体解的存在性

The Global Existence of Solutions to a Class of Two-Dimensional Keller-Segel Systems

摘要: 该文研究一类Keller-Segel方程组的Cauchy问题：$\{\begin{array}{ll}{u}_t=div\left(\nabla u-\chi u\nabla v\right),\hfill & x\in {ℝ}^2,t\in \left(0,\infty \right),\hfill \\ \kappa v_t=\Delta v-\lambda v+u,\hfill & x\in {ℝ}^2,t\in \left(0,\infty \right),\hfill \\ u\left(x,0\right)=u_0\left(x\right)\ge 0,v\left(x,0\right)=v_0\left(x\right)\ge 0,\hfill & x\in {ℝ}^2,\hfill \end{array}$ 其中$\kappa \ge 0$ ，$\chi >1$ ，$\lambda >0$ ，且$u_0\in L^1\left({ℝ}^2\right)\cap L^{\infty }\left({ℝ}^2\right),v_0\in L^1\left({ℝ}^2\right)\cap H^1\left({ℝ}^2\right)$ ，通过构造更一般的能量泛函和能量不等式，证明得到若细胞或细菌的初始质量满足${\displaystyle {\int }_{{ℝ}^2}{u}_0}dx<min\left\{\frac{8\pi }{\chi },\frac{1}{2\left(\frac{\kappa {\left(\sqrt{\chi }-1\right)}^2}{4{\epsilon }^{*}}+{\left(\sqrt{\chi }-1\right)}^2\right)max\left\{\frac{C_{GN}}{2},9\right\}}\right\},$ 其中${\epsilon }^{*}\in \left(0,\frac{1}{2{\left(\sqrt{\chi }-1\right)}^2}\right)$ ，$C_{GN}$ 为Gagliardo-Nirenberg常数，则该模型的解整体存在。

Abstract: This paper investigates a class of Keller-Segel systems described by the Cauchy problem: $\{\begin{array}{ll}{u}_t=div\left(\nabla u-\chi u\nabla v\right),\hfill & x\in {ℝ}^2,t\in \left(0,\infty \right),\hfill \\ \kappa v_t=\Delta v-\lambda v+u,\hfill & x\in {ℝ}^2,t\in \left(0,\infty \right),\hfill \\ u\left(x,0\right)=u_0\left(x\right)\ge 0,v\left(x,0\right)=v_0\left(x\right)\ge 0,\hfill & x\in {ℝ}^2,\hfill \end{array}$ where $\kappa \ge 0$ , $\chi >1$ , and $\lambda >0$ . By constructing a more general energy functional together with a corresponding energy inequality, we prove global-in-time existence provided the initial cell (bacterial) mass satisfies ${\displaystyle {\int }_{{ℝ}^2}{u}_0}dx<min\left\{\frac{8\pi }{\chi },\frac{1}{2\left(\frac{\kappa {\left(\sqrt{\chi }-1\right)}^2}{4{\epsilon }^{*}}+{\left(\sqrt{\chi }-1\right)}^2\right)max\left\{\frac{C_{GN}}{2},9\right\}}\right\},$ for some ${\epsilon }^{*}\in \left(0,\frac{1}{2{\left(\sqrt{\chi }-1\right)}^2}\right)$ . Here $C_{GN}$ denotes the Gagliardo-Nirenberg constant. Under this smallness condition on the initial mass, the model admits a global solution. Then the global existence of classical solutions to this system can be established.