Spatial Decay of Rotating Waves in Reaction Diffusion Systems

Abstract

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 $f:{\mathbb{R}}^N\to {\mathbb{R}}^N$ 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 $W_{\theta }^{1,p}({\mathbb{R}}^d,{\mathbb{R}}^N)$. 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 $\mathrm{Re}\lambda$ 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.

44 pages, 15 figures