PUB - Publikationen an der Universität Bielefeld

In this paper we study nonlinear problems for Ornstein-Uhlenbeck operators A Delta v(x) + < Sx, del v(x)> + f(v(x)) = 0, x is an element of R-d, d >= 2, where the matrix A is an element of R-N,R- N is diagonalizable and has eigenvalues with positive real part, the map f : R-N -> R-N is sufficiently smooth and the matrix S is an element of R-d,R- 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(*) 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-theta(1, p) (R-d, 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 eigen-value problem A Delta v(x) + < Sx, del v(x)> + Df(v(*)(x))v(x) = lambda v(x), x is an element of R-d, d >= 2, decays exponentially in space, provided 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.