In this paper, we propose and investigate a memory-based reaction-diffusion equation with nonlocal maturation delay and homogeneous Dirichlet boundary condition. We first study the existence of the spatially inhomogeneous steady state. By analyzing the associated characteristic equation, we obtain sufficient conditions for local stability and Hopf bifurcation of this inhomogeneous steady state, respectively. For the Hopf bifurcation analysis, a geometric method and prior estimation techniques are combined to find all bifurcation values because the characteristic equation includes a non-self-adjoint operator and two time delays. In addition, we provide an explicit formula to determine the crossing direction of the purely imaginary eigenvalues. The bifurcation analysis reveals that the diffusion with memory effect could induce spatiotemporal patterns which were never possessed by an equation without memory-based diffusion. Furthermore, these patterns are different from the ones of a spatial memory equation with Neumann boundary condition.

中文翻译：

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具有非局部成熟延迟和敌对边界条件的空间记忆模型的分析

在本文中，我们提出并研究了基于记忆的具有非局部成熟延迟和齐次Dirichlet边界条件的反应扩散方程。我们首先研究空间非均匀稳态的存在。通过分析相关的特征方程，我们分别获得了该非均匀稳态的局部稳定性和Hopf分支的充分条件。对于霍普夫分叉分析，由于特征方程包括一个非自伴算子和两个时间延迟，因此将几何方法和先验估计技术结合起来可以找到所有分叉值。此外，我们提供了一个明确的公式来确定纯虚数特征值的交叉方向。分叉分析表明，具有记忆效应的扩散可能会引起时空模式，而没有基于记忆的扩散方程永远不会拥有这种时空模式。此外，这些模式不同于带有Neumann边界条件的空间记忆方程式。