Mixed problem with integral boundary condition for a high order mixed type partial differential equation (Q1812270)

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 \[ {\partial^2 u\over\partial t^2}+ (-1)^\alpha {\partial^\alpha\over\partial x^\alpha} \Biggl(a(x, t){\partial^{\alpha+1} u\over\partial x^\alpha\partial t}\Biggr)= f(x,t) \] in the rectangle \(Q= (0,1)\times (0,T)\), where \(a(x,t)\) is bounded for a \(0< a_0< a(x, t)\leq a_1\), and has bounded partial derivatives such that \(0< a_2\leq{\partial(x,t)\over\partial t}\leq a_3\) and \({\partial^i a(x,t)\over\partial x^i}\leq b_i\), \(i= 1,\dots,\alpha\) for \((x,t)\in\overline Q\). To this equation they add the initial conditions \[ (x,0)= \varphi(x),\quad {\partial u\over\partial t} (x,0)= \psi(x), \quad x\in (0,1), \] the boundary conditions \({\partial^i\over\partial x^i} u(0,t)= 0\) for \(0\leq i\leq\alpha-1\), \(t\in (0,T)\), \({\partial^i\over\partial x^i} u(1,t)= 0\) for \(0\leq i\leq\alpha- 2\), \(t\in (0,T)\), and integral condition \[ \int^1_0 u(\xi, t)\,d\xi= 0,\quad t\in (0,T), \] 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.