Space Almost Periodic Solutions of Reaction Diffusion Equations

Abstract

We consider reaction diffusion equations of the form (*) ∂ _t u = νΔu + ζ u + $\varsigma u + \mathcal{P}\left( u \right),\mathcal{P}\left( u \right) = \sum _z^m a_k u^k$ and seek solutions on ℝⁿ which are almost periodic in the space variables x. Such solutions are constructed in the space H ⁰(ℝⁿ) of almost periodic functions f(x) subject to (**) $f\left( x \right) = \sum f_k e^{i\nabla _k x} ,\sum \left| {fk} \right| < \infty$ , provided that the coefficients a _k in (*) are also in this class. Such solutions are obtained via an instable manifold construction, which yields solutions on t ∈ (− ∞, 0] of slow exponential decay. An extension of the method to Fourier transforms of complex measures is outlined.