Global dynamics of delayed reaction–diffusion equations in unbounded domains
Global dynamics of delayed reaction–diffusion equations in unbounded domains
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DOI：
10.1007/s00033-012-0224-x
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发表时间：
2012-05
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影响因子：
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通讯作者：
Taishan Yi;Yuming Chen;Jianhong Wu
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文献类型：
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作者：
Taishan Yi;Yuming Chen;Jianhong Wu
我们考虑无界域中的非局部延迟反应扩散方程，其中包括由种群动态引起的一些特殊情况。由于空间域的非紧性，解半流不是紧的。我们首先证明，对于自然相空间的紧致开放拓扑，这些解在有界且正不变的集合 YinC+=C([−1, 0],X+) 上引出紧致且连续的半流，它吸引方程的每个解，其中 X+ 是从 [0, ∞) 开始的所有有界且一致连续函数的集合。然后，为了克服描述全局动力学的困难，我们在描述非局部延迟效应和扩散算子的微妙渐近性质后，建立了非平凡解的先验估计。该估计使我们能够显示方程相对于紧凑开放拓扑的持久性。借助永久性，我们可以采用标准动力系统理论论证来建立非平凡均衡的全局吸引力。主要结果用扩散 Nicholson 的绿蝇方程和扩散 Mackey-Glass 方程来说明。
We consider a nonlocal delayed reaction–diffusion equation in an unbounded domain that includes some special cases arising from population dynamics. Due to the non-compactness of the spatial domain, the solution semiflow is not compact. We first show that, with respect to the compact open topology for the natural phase space, the solutions induce a compact and continuous semiflowon a bounded and positively invariant setYinC+=C([−1, 0],X+) that attracts every solution of the equation, whereX+is the set of all bounded and uniformly continuous functions fromto [0, ∞). Then, to overcome the difficulty in describing the global dynamics, we establish a priori estimate for nontrivial solutions after describing the delicate asymptotic properties of the nonlocal delayed effect and the diffusion operator. The estimate enables us to show the permanence of the equation with respect to the compact open topology. With the help of the permanence, we can employ standard dynamical system theoretical arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated with the diffusive Nicholson’s blowfly equation and the diffusive Mackey–Glass equation.