In this paper, we prove a Hopf bifurcation theorem for second-order semi-linear equations involving non-densely defined operators. Here, we use the Crandall and Rabinowitz’s approach based on a suitable application of the implicit function theorem. As a special case, we obtain the existence of periodic wave trains for the so-called Gurtin–MacCamy problem arising in population dynamics and that couples both spatial diffusion and age structure.

中文翻译：

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二阶半线性 Gurtin-MacCamy 方程的 Hopf 分岔定理

在本文中，我们证明了涉及非密集定义算子的二阶半线性方程的 Hopf 分岔定理。在这里，我们使用基于隐函数定理的适当应用的 Crandall 和 Rabinowitz 方法。作为一个特例，我们为人口动态中出现的所谓 Gurtin-MacCamy 问题获得了周期性波列的存在，该问题结合了空间扩散和年龄结构。