Degenerating parabolic equations with a variable direction of evolution

The aim of the paper is to study the solvability in the classes of regular solutions of boundary value problems for differential equations
$$ \varphi(t)u_t-\psi(t)\Delta u+c(x,t)u=f(x,t)\quad (x\in\Omega\subset \mathbb{R}^n,\quad 0<t<T). $$
A feature of these equations is that the function $\varphi (t)$ in them can arbitrarily change the sign on the segment $[0, T]$, while the function $\psi (t)$ is nonnegative for $t \in [0, T]$. For the problems under consideration, we prove existence and uniqueness theorems.