L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves
Assuming spectral stability conditions of periodic reaction-diffusion waves u¯(x), we consider L1(R)-nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initial modulations. Lp(R)-nonlinear...
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Wiley
2017-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2017/3824501 |
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| author | Soyeun Jung |
| author_facet | Soyeun Jung |
| author_sort | Soyeun Jung |
| collection | DOAJ |
| description | Assuming spectral stability conditions of periodic reaction-diffusion waves u¯(x), we consider L1(R)-nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initial modulations. Lp(R)-nonlinear stability of such waves has been studied in Johnson et al. (2013) for p≥2 by using Hausdorff-Young inequality. In this note, by using the pointwise estimates obtained in Jung, (2012) and Jung and Zumbrun (2016), we extend Lp(R)-nonlinear stability (p≥2) in Johnson et al. (2013) to L1(R)-nonlinear stability. More precisely, we obtain L1(R)-estimates of modulated perturbations u~(x-ψ(x,t),t)-u¯(x) of u¯ with a phase function ψ(x,t) under small initial perturbations consisting of localized initial perturbations u~(x-h0(x),0)-u¯(x) and nonlocalized initial modulations h0(x)=ψ(x,0). |
| format | Article |
| id | doaj-art-a21a1e84bdc7462f99acdd837de7ff9c |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
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| series | Advances in Mathematical Physics |
| spelling | doaj-art-a21a1e84bdc7462f99acdd837de7ff9c2025-02-03T01:26:42ZengWileyAdvances in Mathematical Physics1687-91201687-91392017-01-01201710.1155/2017/38245013824501L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion WavesSoyeun Jung0Kongju National University, Gongju-si, Chungcheongnam-do 314-701, Republic of KoreaAssuming spectral stability conditions of periodic reaction-diffusion waves u¯(x), we consider L1(R)-nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initial modulations. Lp(R)-nonlinear stability of such waves has been studied in Johnson et al. (2013) for p≥2 by using Hausdorff-Young inequality. In this note, by using the pointwise estimates obtained in Jung, (2012) and Jung and Zumbrun (2016), we extend Lp(R)-nonlinear stability (p≥2) in Johnson et al. (2013) to L1(R)-nonlinear stability. More precisely, we obtain L1(R)-estimates of modulated perturbations u~(x-ψ(x,t),t)-u¯(x) of u¯ with a phase function ψ(x,t) under small initial perturbations consisting of localized initial perturbations u~(x-h0(x),0)-u¯(x) and nonlocalized initial modulations h0(x)=ψ(x,0).http://dx.doi.org/10.1155/2017/3824501 |
| spellingShingle | Soyeun Jung L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves Advances in Mathematical Physics |
| title | L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves |
| title_full | L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves |
| title_fullStr | L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves |
| title_full_unstemmed | L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves |
| title_short | L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves |
| title_sort | l1 r nonlinear stability of nonlocalized modulated periodic reaction diffusion waves |
| url | http://dx.doi.org/10.1155/2017/3824501 |
| work_keys_str_mv | AT soyeunjung l1rnonlinearstabilityofnonlocalizedmodulatedperiodicreactiondiffusionwaves |