Space Almost Periodic Solutions of Reaction Diffusion Equations
Bruno Scarpellini · Birkhäuser Basel eBooks · 2007
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 ℝ n which are almost periodic in the space variables x. Such solutions are constructed in the space H 0(ℝ n ) of almost periodic functions f(x) subject to (**) $$ f\left( x \right) = \sum f_k e^{i abla _k x} ,\sum \left| {fk} \right| < \infty $$ , 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.