@Article{NMTMA-15-938,
author = {Gomez , DanielWei , Jun-Cheng and Yang , Wen},
title = {Stability of Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2022},
volume = {15},
number = {4},
pages = {938--989},
abstract = {<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>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2022-0003s},
url = {http://global-sci.org/intro/article_detail/nmtma/21086.html}
}