A nonlocal boundary value problem for a class of Petrovskii well-posed equations

As is well known, the mixed problem for the entire class of Petrovskii well-posed partial differential equations has not been studied. In this paper, a certain subclass of Petrovskii well-posed equations for which it is possible to state and study mixed problems, is isolated. In the rectangle $[0,T]\times[0,1]$, consider the equation
$$ D_t^2u+aD_tD_x^{2k}u+bD_x^{2p}u+\sum\limits_{\alpha\leqslant{2k-1}} a_\alpha(t,x)D_tD_x^\alpha+\sum\limits_{\alpha\leqslant{2p-1}}b_\alpha(t,x)D_x^\alpha u=f(t, x) $$
with boundary conditions
$$ L_\nu u=\alpha_\nu u_x^{(q_\nu)}(t,0)+\beta_\nu u_x^{(q_\nu)}(t,1)+ T_\nu u(t,\cdot)=0, \qquad \nu=1\div2k, $$
for $p\leqslant k$, where $|\alpha_\nu|+|\beta_\nu|\ne 0$, $\nu=1\div2k$, $0\leqslant q_\nu\leqslant q_{\nu+1}$, $q_\nu<q_{\nu+2}$, $T_\nu$ is a continuous linear functional in $W_q^{q_\nu}(0, 1)$, $q<+\infty$, and for $k<p<2k$
$$ L_{2k+s}u=L_{n_s}u^{(2k)}=\alpha_{n_s}u_x^{(q_{n_s}+2k)}(t,0)+ \beta_{n_s}u_x^{(q_{n_s}+2k)}(t,1)+T_{n_s}u_x^{(2k)}(t,\cdot)=0, $$
$s=1\div2p-2k$, $1\leqslant n_s\leqslant2k$, and with initial conditions $u(0,x)=u_0(x)$ and $u'_t(0,x)=u_1(x)$.
Well-posedness conditions are found for this problem.
Bibliography: 9 titles.