Spatial decay of rotating waves in reaction diffusion systems

Beyn W-J, Otten D (2016)
Dynamics of Partial Differential Equations 13(3): 191-240.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
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.
Stichworte
Rotating waves; spatial exponential decay; Ornstein-Uhlenbeck operator; exponentially weighted resolvent estimates; reaction-diffusion equations
Erscheinungsjahr
2016
Zeitschriftentitel
Dynamics of Partial Differential Equations
Band
13
Ausgabe
3
Seite(n)
191-240
ISSN
1548-159X
Page URI
https://pub.uni-bielefeld.de/record/2905140

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Beyn W-J, Otten D. Spatial decay of rotating waves in reaction diffusion systems. Dynamics of Partial Differential Equations. 2016;13(3):191-240.
Beyn, W. - J., & Otten, D. (2016). Spatial decay of rotating waves in reaction diffusion systems. Dynamics of Partial Differential Equations, 13(3), 191-240. doi:10.4310/DPDE.2016.v13.n3.a2
Beyn, W. - J., and Otten, D. (2016). Spatial decay of rotating waves in reaction diffusion systems. Dynamics of Partial Differential Equations 13, 191-240.
Beyn, W.-J., & Otten, D., 2016. Spatial decay of rotating waves in reaction diffusion systems. Dynamics of Partial Differential Equations, 13(3), p 191-240.
W.-J. Beyn and D. Otten, “Spatial decay of rotating waves in reaction diffusion systems”, Dynamics of Partial Differential Equations, vol. 13, 2016, pp. 191-240.
Beyn, W.-J., Otten, D.: Spatial decay of rotating waves in reaction diffusion systems. Dynamics of Partial Differential Equations. 13, 191-240 (2016).
Beyn, Wolf-Jürgen, and Otten, Denny. “Spatial decay of rotating waves in reaction diffusion systems”. Dynamics of Partial Differential Equations 13.3 (2016): 191-240.