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 $${\partial }_t^{\alpha }u(t,x)=a^{ij}(t,x)u_{x^i{x}^j}(t,x)+f(t,x,u),t>0,x\in {\mathbf{R}}^d.$$ Here, $\alpha \in (0,2)$, $p,q>1$, and ${\partial }_t^{\alpha }$ is the Caupto fractional derivative of order $\alpha$. 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.

References in corpus (2)

• Waiting-times and returns in high-frequency financial data: an empirical study
• Parabolic BMO estimates for pseudo-differential operators of arbitrary order

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