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.

翻译：Gray-Scott模型是一组描述远离平衡态化学系统的反应-扩散方程。该模型之所以引起关注，源于其能够产生包含脉冲、斑点、条纹及自复制模式在内的时空结构。我们考虑了该模型的一种扩展形式，其中不同化学物质的扩散被假定为非局部的，并可通过积分算子表示。特别地，我们重点研究了具有有限二阶矩的严格正、对称的$L^1$卷积核。在有限区间上对方程进行建模，我们证明了在非局部Dirichlet和Neumann边界条件下小时间弱解的存在性。基于此结果，我们开发了一种有限元数值格式，用以探索非局部扩散对脉冲解形成的影响。