*(English)*Zbl 1035.35085

In this paper a mixed problem with integral boundary conditions for a high-order partial differential equation of mixed type is studying. The authors consider the equation

in the rectangle $Q=(0,1)\times (0,T)$, where $a(x,t)$ is bounded for a $0<{a}_{0}<a(x,t)\le {a}_{1}$, and has bounded partial derivatives such that $0<{a}_{2}\le \frac{\partial (x,t)}{\partial t}\le {a}_{3}$ and $\frac{{\partial}^{i}a(x,t)}{\partial {x}^{i}}\le {b}_{i}$, $i=1,\cdots ,\alpha $ for $(x,t)\in \overline{Q}$. To this equation they add the initial conditions

the boundary conditions $\frac{{\partial}^{i}}{\partial {x}^{i}}u(0,t)=0$ for $0\le i\le \alpha -1$, $t\in (0,T)$, $\frac{{\partial}^{i}}{\partial {x}^{i}}u(1,t)=0$ for $0\le i\le \alpha -2$, $t\in (0,T)$, and integral condition

where $\varphi $ and $\psi $ are known functions which satisfy the compatibility conditions given in the last three equations.

The existence and uniqueness of the solution are proved as the proof is based on energy inequality, and on the density of the range of the operator generated by the considered problem.

##### MSC:

35M10 | PDE of mixed type |

35B45 | A priori estimates for solutions of PDE |

35G10 | Initial value problems for linear higher-order PDE |