Solution of a parabolic boundary-value problem in a space of generalized functions with the help of the Green matrix (Q579518)

The domain \(\Omega_ 0\subset R^ n\), bounded by the surface \(\Omega_ 1\) of \(C^{\infty}\) class, the sets \(Q_ i=[0,T]\times \Omega_ i\), \(i=0,1\), and the spaces \(D(\bar Q_ 0)\) and \(D(Q_ 1)\) of functions of \(C^{\infty}\) class being given, the purpose of this paper is to study the boundary value problem: \[ (D_ t-\sum_{| k| \leq 2b}a_ k(t,x)D^ k_ x)u=F_ 0\quad in\quad \bar Q_ 0, \] \[ \sum_{| k| \leq r_ j}b_{jk}(t,x)D_ x^ k u|_{Q_ 1}=F_ j,\quad j=1,2,...,bp=m,\quad r_ j\leq 2b-1;\quad u|_{t=0}=F_{m+1}. \] Here \(a_ k(t,x)\) are quadratic matrices with elements from \(D(\bar Q_ 0)\) and \(b_{jk}(t,x)\) are column vectors with p elements of \(D(Q_ 1)\). Existence and uniqueness theorems are proved.