Spatial Decay of Rotating Waves in Reaction-Diffusion Systems

In this paper we study nonlinear problems for Ornstein-Uhlenbeck operators A△v(x) + 〈Sx,∇v(x)〉+ f(v(x)) = 0, x ∈ R, d > 2, where the matrix A ∈ R is diagonalizable and has eigenvalues with positive real part, the map f : R → R is sufficiently smooth and the matrix S ∈ R 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⋆ of the nonlinear problem, which falls below a certain threshold at infinity, already decays exponentially in space, in the sense that v⋆ belongs to an exponentially weighted Sobolev space W 1,p θ (R ,R ). 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△v(x) + 〈Sx,∇v(x)〉+Df(v⋆(x))v(x) = λv(x), x ∈ R d , d > 2, decays exponentially in space, provided Reλ 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.