Stability of weak solutions of parabolic systems with distributed parameters on the graph

The analysis of the behaviour of the evolutionary equation solution with unlimited time has been a subject of discussion in scientific circles for a long time. There are practical reasons for this when the initial conditions of the equation are specified a certain error: how the small changes in the initial conditions affect the behaviour the solution for large values of the time. The paper uses the classical understanding of stability of the solution of a differential equation or a system of equations that goes to the works of A. M. Lyapunov: a solution is stable if it little changes under the perturbations of the initial condition. In the work specified the stability conditions the solution of an evolutionary parabolic system with distributed parameters on a describing the process of transfer of a continuous mass in a spatial network are. The parabolic system is considered in the weak formulation: a weak solution of system is a summable function that satisfies the integral form identity, which determines variational formulation for the initial-boundary value problem. By going beyond the (smooth) solutions and addressing weak solutions of the problem the authors aim only to describe more precisely the physical nature of the transfer processes (this takes on importance when studying the dynamics of multiphase media) but also to the path processes in multidimensional network-like domains. The used approach is based on priori estimates of the weak solution and the construction (the Fayedo—Galerkin method a special basis — the system of eigenfunctions of the elliptic operator of a parabolic) of a weakly compact family of approximate solutions in the selected state space. obtained results underlie the analysis of optimal control problems for differential systems distributed parameters on a graph, which have interesting analogies with multiphase of multidimensional hydrodynamics.