Dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity
Dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity
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DOI：
10.1017/s0308210509000262
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发表时间：
2010-10
期刊：
影响因子：
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通讯作者：
Zhi-Cheng Wang;Wan-Tong Li
中科院分区：
文献类型：
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作者：
Zhi-Cheng Wang;Wan-Tong Li
本文研究了一类非局部非局部时滞反应扩散方程的动力学性质，该方程可由阶段结构的单种群增长导出。我们首先证明了当初值为非负有界时，柯西型问题的解是正保有界的。然后，通过建立比较定理和一系列比较论证，证明了正平衡点的全局吸引性。当不存在正平衡点时，我们证明了零平衡点是全局吸引的。特别地，这些结果对于具有Neumann边界条件的有界域上的非局部时滞反应扩散方程仍然有效。最后，我们利用两个辅助方程的行波解和正平衡点的全局吸引性证明了新的整体解的存在性。
Abstract This paper is concerned with the dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity on an infinite n-dimensional domain, which can be derived from the growth of a stage-structured single-species population. We first prove that solutions of the Cauchy-type problem are positively preserving and bounded if the initial value is non-negative and bounded. Then, by establishing a comparison theorem and a series of comparison arguments, we prove the global attractivity of the positive equilibrium. When there exist no positive equilibria, we prove that the zero equilibrium is globally attractive. In particular, these results are still valid for the non-local delayed reaction–diffusion equation on a bounded domain with the Neumann boundary condition. Finally, we establish the existence of new entire solutions by using the travelling-wave solutions of two auxiliary equations and the global attractivity of the positive equilibrium.