论文标题
反应 - 扩散系统中周期性波列的非线性稳定性和渐近行为，反应$ c _ {\ mathrm {ub}} $ - 扰动
Nonlinear stability and asymptotic behavior of periodic wave trains in reaction-diffusion systems against $C_{\mathrm{ub}}$-perturbations
论文作者
论文摘要
我们提出了一种非线性稳定性理论，该理论是针对反应扩散系统中的周期性波序列的，该理论仅依赖于纯$ l^\ infty $估计。我们的分析表明，如当前文献所示，可以完全取消对扰动的本地化或周期性要求。受到以前考虑局部扰动的作品的启发，我们分解了由线性化围绕波列的线性化产生的半群，并引入了时空相调制以捕获最关键的动力学，该动力受粘性汉堡方程的控制。然后，我们的目标是通过对相应的Duhamel公式进行迭代估计来结束非线性稳定性论点，在缺乏本地化的情况下，我们必须依靠扩散平滑来使半群的衰减。但是，这种衰减不足以控制Duhamel配方中的所有术语。我们通过应用Cole-HOPF变换来消除关键汉堡的非线性来解决这一困难。最终，我们建立了相对于$ c _ {\ mathrm {ub}} $扰动的频谱稳定波列的非线性稳定性。此外，我们表明，扰动的溶液会收敛到调制波序列，其相位和波数通过对相关的粘性汉密尔顿 - 雅各布和汉堡方程的溶液近似。
We present a nonlinear stability theory for periodic wave trains in reaction-diffusion systems, which relies on pure $L^\infty$-estimates only. Our analysis shows that localization or periodicity requirements on perturbations, as present in the current literature, can be completely lifted. Inspired by previous works considering localized perturbations, we decompose the semigroup generated by the linearization about the wave train and introduce a spatio-temporal phase modulation to capture the most critical dynamics, which is governed by a viscous Burgers' equation. We then aim to close a nonlinear stability argument by iterative estimates on the corresponding Duhamel formulation, where, hampered by the lack of localization, we must rely on diffusive smoothing to render decay of the semigroup. Yet, this decay is not strong enough to control all terms in the Duhamel formulation. We address this difficulty by applying the Cole-Hopf transform to eliminate the critical Burgers'-type nonlinearities. Ultimately, we establish nonlinear stability of diffusively spectrally stable wave trains against $C_{\mathrm{ub}}$-perturbations. Moreover, we show that the perturbed solution converges to a modulated wave train, whose phase and wavenumber are approximated by solutions to the associated viscous Hamilton-Jacobi and Burgers' equation, respectively.
