On integral representation of the solutions of a model $\vec{2b}$-parabolic boundary value problem

Abstract

A general boundary value problem for Eidelman type $\overrightarrow{2b}$-parabolic system of equation without minor terms in the equations and boundary conditions, and with constant coefficients in the group of major terms is considered in the region $$\{(t,x_1,\dots,x_n)\in \mathbb{R}^{n+1}|t\in(0,T], x_j\in\mathbb{R}, j\in\{1,\dots,n-1\}, x_n>0\},$$ $T>0$, $n\ge 2$. It is assumed that the boundary conditions are connected with the system of equations by the complementing condition, which is analogous to the Lopatynsky complementing condition. Integral representations of the solutions for such a problem are derived. The kernels of the integrals from this representation form the Green's matrix of the problem. It is revealed that, in general, not all the elements of the Green's matrix are ordinary functions. Some of them contain terms that are linear combinations of Dirac delta functions and their derivatives. This occurs in cases when the boundary conditions include derivatives with respect to the variables $t$ and $x_n$ of orders that are equal or greater than the highest orders of derivatives with respect to these variables in the equations of the system. The obtained results are important, in particular, for the establishing of the correct solvability and integral representation of solutions for more general $\overrightarrow{2b}$-parabolic boundary value problems.