论文标题
非局部灰-Scott模型的分析和模拟
Analysis and Simulations of a Nonlocal Gray-Scott Model
论文作者
论文摘要
灰色 - 斯科特模型是一组反应扩散方程，它描述了远离平衡的化学系统。对该模型的兴趣源于其产生时空结构的能力，包括脉冲，斑点，条纹和自我复制模式。我们考虑了该模型的扩展，其中假定不同化学物质的传播是非局部性的，因此可以由积分算子代表。特别是，我们专注于严格的积极，对称性，$ l^1 $卷积内核的情况，这些核心具有有限的第二时刻。在有限间隔内对方程进行建模，我们证明了在非本地Dirichlet和Neumann边界约束的情况下存在小型弱解。然后，我们使用此结果来开发有限的元素数值方案，该方案有助于我们探索非局部扩散对脉冲溶液形成的影响。
The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, $L^1$ convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.
