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.
