arXiv 1602.033932016-02-10math.AP

反应扩散系统中旋转波的空间衰减

Spatial Decay of Rotating Waves in Reaction Diffusion Systems

Wolf-Jürgen Beyn, Denny Otten

备注：: 44 pages, 15 figures

摘要：

本文研究Ornstein-Uhlenbeck算子\begin{align*} A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle + f(v(x)) = 0,\,x\in\mathbb{R}^d,\,d\geqslant 2, \end{align*}的非线性问题，其中矩阵$A\in\mathbb{R}^{N,N}$可对角化，特征值为正实部，映射$f:\mathbb{R}^N\rightarrow\mathbb{R}^N$充分光滑，无界漂移项中的矩阵$S\in\mathbb{R}^{d,d}$是反对称的.这种形式的非线性问题表现为含时反应扩散系统中旋转波的定常方程。我们证明在适当的条件下，每一个有界的经典解$v_{\star}$的非线性问题，它在无穷远低于一定的阈值，已经在空间中指数衰减，在这个意义上说，$v_{\star}$属于指数加权Sobolev空间$ W^{1,p}_θ(\mathbb{R}^d,\mathbb{R}^N)$。这个基本结果的几个扩展：复值系统，指数衰减高阶Sobolev空间和逐点估计。我们还证明了特征值问题\begin{align*} A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle + Df(v_{\star}(x))v(x) = λv(x),\,x\in\mathbb{R}^d,\,d\geqslant 2, \end{align*}的每一个有界经典解$v$在空间中指数衰减，只要$\mathrm{Re}\,λ$位于本质谱的右边。作为应用，我们分析了Ginzburg-Landau方程中的自旋孤子解。我们的结果形成的基础上，调查非线性稳定性的旋转波在更高的空间维度和截断有界域。

英文摘要：

In this paper we study nonlinear problems for Ornstein-Uhlenbeck operators \begin{align*} A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle + f(v(x)) = 0,\,x\in\mathbb{R}^d,\,d\geqslant 2, \end{align*} where the matrix $A\in\mathbb{R}^{N,N}$ is diagonalizable and has eigenvalues with positive real part, the map $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is sufficiently smooth and the matrix $S\in\mathbb{R}^{d,d}$ in the unbounded drift term is skew-symmetric. Nonlinear problems of this form appear as stationary equations for rotating waves in time-dependent reaction diffusion systems. We prove under appropriate conditions that every bounded classical solution $v_{\star}$ of the nonlinear problem, which falls below a certain threshold at infinity, already decays exponentially in space, in the sense that $v_{\star}$ belongs to an exponentially weighted Sobolev space $ W^{1,p}_θ(\mathbb{R}^d,\mathbb{R}^N)$. Several extensions of this basic result are presented: to complex-valued systems, to exponential decay in higher order Sobolev spaces and to pointwise estimates. We also prove that every bounded classical solution $v$ of the eigenvalue problem \begin{align*} A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle + Df(v_{\star}(x))v(x) = λv(x),\,x\in\mathbb{R}^d,\,d\geqslant 2, \end{align*} decays exponentially in space, provided $\mathrm{Re}\,λ$ lies to the right of the essential spectrum. As an application we analyze spinning soliton solutions which occur in the Ginzburg-Landau equation. Our results form the basis for investigating nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains.

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