  We establish existence, uniqueness, and arbitrary order Sobolev regularity
results for the second order parabolic equations with measurable coefficients
defined on the conic domains $D$ of the type $$ D(M):=\left\{x\in R^d
:\,\frac{x}{|x|}\in M\right\}, \quad \quad M \subset S^{d-1}. $$ We obtain the
regularity results by using a system of mixed weights consisting of appropriate
powers of the distance to the vertex and of the distance to the boundary. We
also provide the sharp ranges of admissible powers of the distance to the
vertex and to the boundary.
