Many models of high energy physics suggest that the cosmological dark sector consists not just one, but a spectrum ultralight scalar particles with logarithmically distributed masses. To study potential signatures low concentrations axion (also known as fuzzy) matter, we modify Lagrangian perturbation theory (LPT) by distinguishing between trajectories different matter species. We further adapt...

In this paper, we deal with the biharmonic heat equation gradient non-linearity. Under suitable condition of initial datum, show that global unique existence mild solution. The main technique in paper is to use Banach?s fixed point theorem combination Lp-Lq evaluation operator.