Stable solvability in spaces of differentiable functions of some two-dimensional integral equations of heat conduction with an operator-semigroup kernel

In this paper, we study two-dimensional vector boundary Fredholm integral equations of the second kind with an operator kernel expressed in terms of a spatial-temporal $C_0$-semigroup. Such two-dimensional integral equations allow one to obtain solutions of vector boundary value problems of the first, second, and third kind for linear differential-operator equations $\Delta_2u=Bu$ in a planar bounded simply connected domain $\Omega^+$ or its exterior $\Omega^-\equiv R^2\setminus\overline{\Omega^+}$. The operator coefficient $\mathbf{B}$ is a generator of the $C_0$-semigroup in space $L_2(I_Y\times I_T)$ ($I_Y\equiv[0, Y]$, $I_T\equiv[0, T]$). In turn, these boundary value problems are possible formulations of initial boundary value problems of heat conduction on the time interval $I_T$ in a homogeneous cylinder $\Omega^+\times I_Y$ or $\Omega^-\times I_Y$ with inhomogeneous boundary conditions of the first, second, and third kind on the lateral surface of the cylinder, zero boundary conditions of the first, second, or third kind (depending on the operator $\mathbf{B}$) on the cylinder bases and zero initial conditions. The main result of this paper is as follows: under condition $\partial\Omega\in C^{k+2}$, the spaces $C^k(\partial\Omega, H_{B}^n(I_Y\times I_T))$ are invariant with respect to direct and inverse operators of the integral equations, and such operators are bounded in these spaces. Here, $C^k(\partial\Omega, H_{B}^n(I_Y\times I_T))$ is the space of vector functions, $k$ times continuously differentiable on the border $\partial\Omega$ with values in the Sobolev type space $H_{B}^n(I_Y\times I_T)$ defined by powers $n+1$ of the operator $B$.