In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions.In particular we rigorously prove the existence of symmetric an...In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions.In particular we rigorously prove the existence of symmetric and asymmetric twospike solutions using a Lyapunov-Schmidt reduction.The linear stability of these two-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain 2×2 matrix.Our rigorous results are complemented by formal calculations of N-spike solutions using the method of matched asymptotic expansions.In addition,we explicitly consider examples of one-and two-spike solutions for which we numerically calculate their relevant existence and stability thresholds.By considering a one-spike solution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectively destabilize or stabilize a single spike solution with respect to oscillatory instabilities.Furthermore,when considering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutions exist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expanded with the introduction of fractional inhibitor diffusivity.However our calculations indicate that asymmetric two-spike solutions are always linearly unstable.展开更多
文摘In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions.In particular we rigorously prove the existence of symmetric and asymmetric twospike solutions using a Lyapunov-Schmidt reduction.The linear stability of these two-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain 2×2 matrix.Our rigorous results are complemented by formal calculations of N-spike solutions using the method of matched asymptotic expansions.In addition,we explicitly consider examples of one-and two-spike solutions for which we numerically calculate their relevant existence and stability thresholds.By considering a one-spike solution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectively destabilize or stabilize a single spike solution with respect to oscillatory instabilities.Furthermore,when considering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutions exist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expanded with the introduction of fractional inhibitor diffusivity.However our calculations indicate that asymmetric two-spike solutions are always linearly unstable.
