该文将研究如下问题 $ \left\{\begin{array}{ll} u_{t}=\Delta{(\gamma(v)u)}, & x\in\Omega, t>0, \\ v_{t}=\Delta v+wz, & x\in\Omega, t>0, \\ w_{t}=-wz, & x\in\Omega, t>0, \\ z_{t}=\Delta z+ u- z, & x\in\Omega, t>0, \\ \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=\frac{\partial z}{\partial \nu}=0, & x\in\partial\Omega, t>0, \\ (u, v, w, z)(x, 0)=(u_0, v_0, w_0, z_0)(x), &x\in\Omega, \end{array}\right. $ 其中, 有界区域$\Omega\subset\mathbb{R} ^n(1\leqq n\leqq 5)$具有光滑边界, $\nu$表示$\partial \Omega$的外法向量, 且$0<\gamma(v)\in C^3[0, \infty)$.在合适的初始条件下, 该文分两种情形来研究模型的全局经典解的存在性与有界性:$ \bullet\; 1\leq n\leq 3$;$ \bullet \;4\leq n\leq 5$, $\gamma_2\geq \gamma(v)\geq \gamma_1>0$, 且$\left|\gamma'(v)\right|\leq \gamma_3 $, $v \in [0, \infty)$, 其中常数$\gamma_i>0\ (i=1, 2, 3)$.接着, 该文计算得到当$t\rightarrow\infty$时, 其解$(u, v, w, z)$将指数收敛到平衡点$(\bar{u}_0, \bar{v}_0+\bar{w}_0, 0, \bar{u}_0)$, 其中$\bar{u}_0=\frac{1}{\left|\Omega\right|}\int_{\Omega}u_0{\rm d}x$, $\bar{v}_0=\frac{1}{\left|\Omega\right|}\int_{\Omega}v_0{\rm d}x$, $\bar{w}_0=\frac{1}{\left|\Omega\right|}\int_{\Omega}w_0{\rm d}x$.
《数学物理学报》诚聘编辑1名