TY - JOUR
T1 - Stability of Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain
AU - Gomez , Daniel
AU - Wei , Jun-Cheng
AU - Yang , Wen
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 938
EP - 989
PY - 2022
DA - 2022/10
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2022-0003s
UR - https://global-sci.org/intro/article_detail/nmtma/21086.html
KW - Gierer-Meinhardt system, eigenvalue, stability, fractional Laplacian, localized solution.
AB - <p style="text-align: justify;">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 two-spike 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.</p>