On Decay of Entropy Solutions to Nonlinear Degenerate Parabolic Equation with Almost Periodic Initial Data

We study the Cauchy problem for nonlinear degenerate parabolic equations with almost periodic initial data. Existence and uniqueness (in the Besicovitch space) of entropy solutions are established. It is demonstrated that the entropy solution remains to be spatially almost periodic and that its spectrum (more precisely, the additive group generated by the spectrum) does not increase in the time variable. Under a precise nonlinearity-diffusivity condition on the input data we establish the long time decay property in the Besicovitch norm. For the proof we use reduction to the periodic case and ergodic methods.