Spatial decay of rotating waves in reaction diffusion systems

In this paper we study nonlinear problems for Ornstein–Uhlenbeck operators\[A \Delta v(x) + \langle Sx, \nabla v(x) \rangle + f(v(x)) = 0, x \in \mathbb{R}^d , d \geqslant 2 \; \textrm{,}\]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 $W^{1,p}_{\theta} (\mathbb{R}^d, \mathbb{R}^N)$. 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\[A \Delta v(x) + \langle Sx, \nabla v(x) \rangle + Df(v_{\star}(x))v(x) =\lambda v(x), x \in \mathbb{R}^d, d \geqslant 2 \; \textrm{,}\]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.

Keywords

rotating waves, spatial exponential decay, Ornstein–Uhlenbeck operator, exponentially weighted resolvent estimates, reaction-diffusion equations

2010 Mathematics Subject Classification

Primary 35K57. Secondary 35B40, 35Pxx, 35Q56, 47A55, 47N40.

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Published 23 June 2016