Influence of parameters on properties of solutions to intergral boundary value problems in a layer (Q1901953)

The present article is devoted to the study of the following boundary value problem in a layer \(\Pi(T)= \mathbb{R}^n\times [0, T]\): \[ \partial u(x, t)/\partial t= P(- iD_x) u(x, t),\quad (x, t)\in \Pi(T),\tag{1} \] \[ \int^T_0 Q(- iD_x) u(x, t)dt= u_0(x),\quad x\in \mathbb{R}^n,\tag{2} \] where both \(P(\sigma)\) and \(Q(\sigma)\) are arbitrary polynomials with complex coefficients \(\sigma\in \mathbb{R}^n\), \(T\in \mathbb{R}_+= ]0, +\infty[\).The aim of the present article is the study of influence of parameters of the problem (1)--(2) (i.e., of coefficients of differential operators \(P(- iD_x)\) and \(Q(- iD_x)\), and of thickness \(T\) of the layer) both on the correctness of the problem on the whole and on properties of its solution.