STABLE AND UNSTABLE INVARIANT SPACES OF ONE STOCHASTIC NON-CLASSICAL EQUATION WITH A RELATIVELY RADIAL OPERATOR ON A 3-TORUS

Abstract

In this paper, we consider a stochastic analogue of the Dzektser equation, which is a model of the evolution of the free surface of a filtered fluid in the spaces of differential forms defined on a smooth compact oriented manifold without boundary. We consider a three-dimensional torus (3-torus) as such a manifold. Also, we consider the question of the stability of solutions to the Dzektser equation in the spaces of "noises" on this manifold in terms of invariant spaces. To this end, the stochastic Dzektser equation is reduced to a linear stochastic Sobolev type equation. We show the existence of stable and unstable invariant spaces and dichotomies of solutions to the stochastic Dzektser equation on a three-dimensional torus. A computational experiment is carried out. An algorithm is developed in the form of a program in the Maple environment. As a result of the implementation of this algorithm, we obtain the following. First, we construct a graph of solutions when the coefficients of the Dzektser equation satisfy sufficient conditions for the existence of only a stable invariant space of this equation. Second, we construct graphs of solutions in the case of the existence of exponential dichotomies of solutions. In this case, we show that the space of solutions splits into stable and unstable invariant spaces such that solutions increase in one of the spaces and decrease in another space.