On a nonlocal two-point boundary value problem in a band for an equation with nonconstant coefficients
The following problem is studied: \[ \frac{\partial u}{\partial t}(x,t)=P(D_x,t)u(x,t)+f(x,t),\;\;(x,t)\in\Pi(T)=\mathbb{R}^n\times(0,T),\eqno{(1)} \] \[ A(D_x)u(x,0)+B(D_x)u(x,T)=u_0(x),\;\;x\in\mathbb{R}^n,\eqno{(2)} \] where \(D_x=(-i\frac{\partial}{\partial x_1},\ldots,-i \frac{\partial}{\partial x_n})\) and \(P(\sigma,t)\), \(A(\sigma)\), and \(B(\sigma)\) are polynomials in \(\sigma\). To study well-posedness of the problem, the author uses the Fourier transform, the Tarskii--Seidenberg theorem, and some methods presented, for instance, in \textit{L. Hörmander}'s monographs [The analysis of linear partial differential operators, vols. I--V (Berlin 1990; Zbl 0712.35001, Zbl 0687.35002, Zbl 0601.35001, Zbl 0612.35001)]. Solvability and uniqueness conditions are given in terms of the behavior of some polynomials in the corresponding parameters \(\sigma\) and \(t\) (\(-i \frac{\partial}{\partial x_k} \rightarrow \sigma_k\)). Weighted \(C^k\)-spaces are used; a solution the author is looking for is of polynomial growth at infinity with respect to \(x\).
scientific article; zbMATH DE number 1037974
scientific article