Boundary value problems for twice degenerate differential equations with multiple characteristics

We study the solvability of boundary value problems for degenerate differential equations of the form
$$ \varphi(t)u_t-(-1)^m\psi(t)D^{2m+1}_{x}u+c(x,t)u=f(x,t) $$
($D^k_x=\frac{\partial^k}{\partial x^k}$, $m\ge1$ is an integer, $x\in(0,1)$, $t\in(0,T)$, $0<T<+\infty$), called equations with multiple characteristics. In these equations, the function $\varphi(t)$ can change the sign on the interval $[0,T]$ arbitrarily, while the function $\psi(t)$ is assumed nonnegative. For the equations under consideration, we propose the formulation of boundary value problems which are essentially determined by numbers $\varphi(0)$ and $\psi(T)$. Existence and uniqueness theorems are proved for the regular solutions that have all Sobolev generalized derivatives entering into the equation.