Unique solvability of parabolic equations with almost periodic coefficients in Hölder spaces. (Q1889488)

Let \(P = {\partial \over \partial t} + {\sum_{| \alpha| \leq 2m}} a_\alpha(x,t)D^\alpha\) be a linear parabolic operator. Assume that the coefficients \(a_\alpha\) are Hölder continuous and almost periodic in the variables \((x,t)\). Consider the problem \(Pu = f\). Indicate with \(Q(P)\) the class of operators of the form \(Q = {\partial \over \partial t} + {\sum_{| \alpha| \leq 2m}} b_\alpha(x,t)D^\alpha\), such that the coefficients \(b_\alpha(x,t)\) are uniform limits of sequences of the form \(a_\alpha(x+\xi_k,t+\tau_k)\) (same sequence \((\xi_k,\tau_k)\) for every \(\alpha\)). The main result of the paper states that this problem has a unique solution \(u\) which is bounded in \(\mathbb R^{n+1}\) for every \(f \in C^{\gamma,\gamma/(2m)}\) (\(0 < \gamma < 1\)), if and only if, for every \(Q \in H(P)\), the problem \(Qv = 0\) has the trivial solution as unique bounded solution in the whole \(\mathbb R^{n+1}\). In this case, \(u\) belongs to the class \(C^{2m,1,\gamma,\gamma/(2m)}\) and is almost periodic if \(f\) is almost periodic. This result is connected to the notion of exponential dichotomy and extensions to other classes of data are considered. Finally, the author examines some quasilinear perturbations of the basic operator.