An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients

Abstract

We introduce an $L_q(L_p)$-theory for the quasi-linear fractional equations of the type $${\partial }^{\hat{I}}{\pm }_{t}u(t,x)=a^{ij}(t,x)u_{x^i{x}^j}(t,x)+f(t,x,u),t>0,x\in {\mathbf{R}}^d.$$ Here, $\hat{I}\pm \in (0,2)$, $p,q>1$, and ${\partial }^{\hat{I}}{\pm }_t$ is the Caupto fractional derivative of order $\hat{I}\pm$. Uniqueness, existence, and $L_q(L_p)$-estimates of solutions are obtained. The leading coefficients $a^{ij}(t,x)$ are assumed to be piecewise continuous in $t$ and uniformly continuous in $x$. In particular $a^{ij}(t,x)$ are allowed to be discontinuous with respect to the time variable. Our approach is based on classical tools in PDE theories such as the Marcinkiewicz interpolation theorem, the Calderon-Zygmund theorem, and perturbation arguments.