## Mixed problem with boundary integral conditions for a certain parabolic equation.(English)Zbl 0864.35049

In the rectangle $$Q=(0,b)\times (0,T)$$, we consider the equation ${{\partial u}\over{\partial t}}+(-1)^m a(t) {{\partial^{2m}u}\over{\partial x^{2m}}}= f(x,t),\tag{1}$ where $$a(t)$$ is bounded, $$0<a_0\leq a(t)\leq a_1$$, and $$a(t)$$ has the bounded derivative such that $$0<c_0\leq a'(t)\leq c_1$$ for $$t\in[0,T]$$. We adhere to equation (1) the initial condition $$u(x,0)=\varphi(x)$$ and the boundary integral conditions $\int^b_0 x^k\cdot u(x,t)dx=0, \qquad k=\overline{0,2m-1}.$ In this paper, the existence and uniqueness of a solution is proved. The proof is based on the method of energy inequalities, presented in [N.-E. Benuar and N. I. Yurchuk, Differ. Uravn. 27, No. 12, 2094-2098 (1991; Zbl 0771.35025)].

### MSC:

 35K35 Initial-boundary value problems for higher-order parabolic equations

### Keywords:

boundary integral conditions; energy inequalities

Zbl 0771.35025
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