Degeneration in differential equations with multiple characteristics

We study the solvability of boundary value problems for the differential equations
$$ \varphi(t)u_t+(-1)^m\psi(t)D^{2m+1}_{x}u+c(x,t)u=f(x,t),\\ \varphi(t)u_{tt}+(-1)^{m+1}\psi(t)D^{2m+1}_{x}u+c(x,t)u=f(x,t), $$
where $x\in(0,1)$, $t\in(0,T),$ $m$ is a non-negative integer, $D^k_x=\frac{\partial^k}{\partial x^k}$ ($D^1_x=D_x$), while the functions $\varphi(t)$ and $\psi(t)$ are non-negative and vanish at some points of the segment $[0,T]$. We prove the existence and uniqueness theorems for the regular solutions, those having all generalized Sobolev derivatives required in the equation, in the inner subdomains.