Mixed problem with boundary integral conditions for a certain parabolic equation
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 [\textit{N.-E. Benuar} and \textit{N. I. Yurchuk}, Differ. Uravn. 27, No. 12, 2094-2098 (1991; Zbl 0771.35025)].
scientific article; zbMATH DE number 947015