2016/02/10 by Wolf-Jürgen Beyn, Beyn, Wolf-Jürgen, Denny Otten +1
Mathematics · #35K57 (35B40 #35Pxx #35Q56 #47A55 #47N40) #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35K57 #msc:35Pxx #msc:35Q56 #msc:47A55
paper · pdf · doi:10.48550/arxiv.1602.03393
44 pages, 15 figures
arxiv created 2016/02/10 · arxiv updated 2016/02/11
In this paper we study nonlinear problems for Ornstein-Uhlenbeck operators A\triangle v(x) + ⟨ Sx,∇ v(x)⟩ + f(v(x)) = 0, x∈ℝd, d\geqslant 2, where the matrix A∈ℝN,N is diagonalizable and has eigenvalues with positive real part, the map f:ℝN→ℝN is sufficiently smooth and the matrix S∈ℝd,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 W1,pθ(ℝd,ℝN). 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) + ⟨ Sx,∇ v(x)⟩ + Df(v⋆(x))v(x) = λv(x), x∈ℝd, d\geqslant 2, decays exponentially in space, provided Re λ 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.