Preprint des Projektes: SFB 701: Spektrale Strukturen und Topologische Methoden in der Mathematik - Projekt B3

Numerische Analyse äquivarianter Evolutionsgleichungen

16-007 Wolf-Jürgen Beyn, Denny Otten.
Spatial Decay of Rotating Waves in Reaction Diffusion Systems

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