Solution of the Cauchy problem with values in refined spaces of Bessel potentials

Abstract

For the Cauchy problem
$$D^{\beta}_t u+a^2(-\Delta)^{\alpha/2}u=F_0(x,t),\;\;\; (x,t) \in \Bbb R^n\times (0,T],$$
$$u(x,0)=F_{1}(x),\;\;\;x\in\Bbb R^n,$$
with the regularized fractional derivative $D^{\beta}_t u$ of order $\beta\in (0,1)$, we establish the existence and uniqueness of the solutions that are classical in time and that take values in certain refined spaces of Bessel potentials.