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 variable has been a subject of discussion in scientiﬁc circles for a long time. There are many practical reasons for this when the initial conditions of the equation are speciﬁed with a certain error: how the small changes in the initial conditions aﬀect the behaviour of the solution for large values of the time. The paper uses the classical understanding of the stability of the solution of a diﬀerential equation or a system of equations that goes back to the works of A. M. Lyapunov: a solution is stable if it little changes under the small perturbations of the initial condition. In the work speciﬁed the stability conditions for the solution of an evolutionary parabolic system with distributed parameters on a graph describing the process of transfer of a continuous mass in a spatial network are indicated. The parabolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisﬁes the integral form identity, which determines the variational formulation for the initial-boundary value problem. By going beyond the classical (smooth) solutions and addressing weak solutions of the problem the authors aim not only to describe more precisely the physical nature of the transfer processes (this takes on particular importance when studying the dynamics of multiphase media) but also to the path analysis processes in multidimensional network-like domains. The used approach is based on a priori estimates of the weak solution and the construction (the Fayedo-Galerkin method with a special basis - the system of eigenfunctions of the elliptic operator of a parabolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems for diﬀerential systems with distributed parameters on a graph, which have interesting analogies with multiphase problems of multidimensional hydrodynamics.