Stability of solutions for systems of delayed parabolic equations

Background. The study is devoted to the analysis of stability in the sense Lyapunov steady state solutions for systems of linear parabolic equations with coefficients depending on time, and with delays depending on time. The cases of continuous and impulsive perturbations are considered. Materials and methods. A method for studying the stability of solutions to systems of linear parabolic equations is as follows. Applying the Fourier transform to the original system of parabolic equations, we arrive at a system of non-stationary ordinary differential equations defined in the spectral region. First, the stability of the resulting system is studied by the method of frozen coefficients in the metric of the space $R_n$ of n-dimensional vectors. Then the resulting statements are extended to the space $L_2$. The application of the Parseval equality allows us to return to the domain of the originals and obtain sufficient conditions for the stability of solutions to systems of linear parabolic equations. Results. An algorithm is proposed that allows one to obtain sufficient stability conditions for solutions of finite systems of linear parabolic equations with time-dependent coefficients and with time-dependent delays. Sufficient stability conditions are expressed in terms of the logarithmic norms of matrices composed of the coefficients of the system of parabolic equations. They are obtained in the metric of the space $L_2$. Algorithms for constructing sufficient stability conditions are efficient, as in the case continuous, and in the case of impulsive perturbations. Conclusions. A method for constructing sufficient stability conditions for solutions of finite systems of linear parabolic equations with time-dependent coefficients and delays. The method can be used in the study non-stationary dynamical systems described by systems of linear parabolic equations with delays depending from time.