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 semilinear fractional equations of the type Here, alpha is an element of (0, 2), p,q > 1, and partial derivative(alpha)(t) 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.