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Mixed problem with integral boundary condition for a high order mixed type partial differential equation. (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

$\frac{{\partial }^{2}u}{\partial {t}^{2}}+{\left(-1\right)}^{\alpha }\frac{{\partial }^{\alpha }}{\partial {x}^{\alpha }}\left(a\left(x,t\right)\frac{{\partial }^{\alpha +1}u}{\partial {x}^{\alpha }\partial t}\right)=f\left(x,t\right)$

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

$\left(x,0\right)=\varphi \left(x\right),\phantom{\rule{1.em}{0ex}}\frac{\partial u}{\partial t}\left(x,0\right)=\psi \left(x\right),\phantom{\rule{1.em}{0ex}}x\in \left(0,1\right),$

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

${\int }_{0}^{1}u\left(\xi ,t\right)\phantom{\rule{0.166667em}{0ex}}d\xi =0,\phantom{\rule{1.em}{0ex}}t\in \left(0,T\right),$

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