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

We introduce an L_q(L_p)-theory for the semilinear fractional equations of the type∂_t ^αu(t,x)=a^ij(t,x)u_xⁱx^j (t,x)+f(t,x,u),t>0,x∈R^d. Here, α∈(0,2), p,q>1, and ∂_t ^α is the Caupto fractional derivative of order α. 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.