Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics - Project B3

Numerical Analysis of equivariant evolution equations

12-139 Denny Otten.
Spatial decay of rotating waves in parabolic systems

In this paper we study solutions of nonlinear systems $A\Delta v (x) + \langle Sx, \nabla v(x)\rangle + f (v(x)) = 0,\quad x \in \mathbb{R}^d,\; d\geq 2.$ The linear operator is of Ornstein-Uhlenbeck type with an unbounded drift term defined by a skew-symmetric matrix $$S\in\mathbb{R}^{d,d}.$$ Equations of this form determine the shape and angular speed of rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that every classical solution which falls below a certain threshold at infinity, must decay exponentially in space. For the proof we utilize the heat kernel matrix of a generalized Ornstein-Uhlenbeck operator, determine its maximal domain and analyze constant and variable coefficient perturbations.