Dynamics of Patterns in Nonlinear Equivariant PDEs

Abstract. In this paper we study nonlinear problems for Ornstein-Uhlenbeck operatorswhere the matrix A ∈ R N,N is diagonalizable and has eigenvalues with positive real part, the map f : R N → R N is sufficiently smooth and the matrix S ∈ 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 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. 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 problemdecays 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.

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“…, m on the imaginary axis. For more details we refer to the examples in Section 6 and to [26,Chapter 9,10].…”

Section: Theorem 51 (Exponential Decay Of Eigenfunctions) (1) Let Tmentioning

confidence: 99%

“…For more details on the computation, see [26,Section 10.3]. Some general theory and applications of the freezing method may be found in [7,8,9,10,37]. Note that in case d = 2 we have a clockwise rotation, if S 12 > 0, and a counter clockwise rotation, if S 12 < 0.…”

Section: Rotating Waves In Reaction Diffusion Systemsmentioning

confidence: 99%