Boundary Value Problems for Some Classes of Singular Parabolic Equations

We study the question of solvability of boundary value problems for the parabolic equation
$$Mu=g(x,t)u_t+L(x,t,D_x)u=f(x,t), \quad (x,t)\in Q=G\times (0,T) \quad (T\le\infty),$$
where $L$ is an elliptic operator in the space variables of order $2m$ defined in a bounded domain $G\subset\mathbb R^n$. We assume that the operator $L$ is coercive and the corresponding boundary value problem $Lu=f$, $B_ju\big|_{\partial G}=0$ admits a variational statement. The function $g(x,t)$ is nonsmooth in $x$ and can change its sign in $Q$.