Existence and stability of symmetric and asymmetric patterns for the half-Laplacian Gray-Scott system in one-dimensional domain

In this paper, we investigate the existence and stability of multiple spikes pattern to the fractional Gray-Scott model with periodic boundary conditions, where the fractional power is $s=\frac{1}{2}$. Employing the classical Lyapunov-Schmidt reduction method, we provide a rigorous proof for the existence of both symmetric multiple spikes and asymmetric two spikes solutions. Furthermore, we analyze the stability of these constructed solutions by studying the associated large and small eigenvalue problems. Our analysis crucially relies on the properties of the Green's function and its derivatives, as well as the study of two nonlocal eigenvalue problems, which play an important role in determining the stability characteristics of the solutions. Moreover, we find out the connection between the eigenvalues of the small eigenvalue problem and the spectral properties of a corresponding circulant matrix. Specially, using the properties of the polygamma function, we establish that all nonzero eigenvalues of this circulant matrix are negative regardless of the number of spikes.