## A strong solution of a high-order mixed type partial differential equation with integral conditions.(English)Zbl 1160.35055

In the rectangle $$Q= (0,1)\times (0,T)$$ the following equation is considered:
${\partial^2u\over\partial t^2}+ (-1)^\alpha a(t){\partial^{2\alpha+1}u\over\partial x^{2\alpha}\partial t}= f(x,t), \tag{1}$
where $$0< a_0< a(t)\leq a_1$$ and $$0< a_2\leq (da(t)/dt)\leq a_3$$ for $$t\in (0,T)$$ and $$\alpha\in \mathbb{N}^*$$. To equation (1) the following conditions are added:
a)
the initial conditions $$u(x, 0)= \varphi(x)$$, $${\partial u\over\partial t} (x, 0)= \psi(x)$$, $$x\in (0,1)$$,
b)
the boundary conditions
\begin{aligned} {\partial^i\over\partial x^i} u(0, t)&= 0\quad\text{for }0\leq i\leq\alpha- k,\quad t\in (0,T),\\ {\partial^i\over\partial x^i} (1,t)&= 0\quad\text{for }0\leq i\leq\alpha- k-1,\quad t\in (0,T),\end{aligned}
and
c)
the integral conditions
$\int^1_0 x^i u(\xi, t)\,d\xi= 0\quad\text{for }0\leq i\leq 2k-2,\quad 1\leq k\leq\alpha,\quad t\in (0,T),$
where $$\varphi$$, $$\psi$$ are known functions, which satisfy compability conditions.
The existence and uniqueness of a strong solution are proved. The proof is based on energy inequality and on the density of the range of the operator generated by the considered problem.

### MSC:

 35M10 PDEs of mixed type 35B45 A priori estimates in context of PDEs 35G10 Initial value problems for linear higher-order PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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