Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2017, Article ID 3824501, 7 pages
Research Article

-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves

Kongju National University, Gongju-si, Chungcheongnam-do 314-701, Republic of Korea

Correspondence should be addressed to Soyeun Jung;

Received 7 February 2017; Revised 31 May 2017; Accepted 10 October 2017; Published 1 November 2017

Academic Editor: Andrei D. Mironov

Copyright © 2017 Soyeun Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Assuming spectral stability conditions of periodic reaction-diffusion waves , we consider -nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initial modulations. -nonlinear stability of such waves has been studied in Johnson et al. (2013) for by using Hausdorff-Young inequality. In this note, by using the pointwise estimates obtained in Jung, (2012) and Jung and Zumbrun (2016), we extend -nonlinear stability () in Johnson et al. (2013) to -nonlinear stability. More precisely, we obtain -estimates of modulated perturbations of with a phase function under small initial perturbations consisting of localized initial perturbations and nonlocalized initial modulations .