SPATIALLY INHOMOGENEOUS SOLUTIONS IN TWO BOUNDARY VALUE PROBLEMS FOR THE CAHN-HILLIARD EQUATIONS

Author(s):  A.N. Kulikov, Dr., Prof., Demidov Yaroslavl State University, Yaroslavl , Russia

D.A. Kulikov, candidate of Sciences, associate Professor, Demidov Yaroslavl State University, Yaroslavl , Russia

Issue:  Volume 51, № 1

Rubric:  Mathematics

Annotation:  The well-known Cahn-Hilliard equation, which has applications in chemical kinetics and physics of boundary phenomena, is considered. This nonlinear differential equation is studied together with homogeneous Dirichlet and Neumann boundary conditions. For both boundary value problems, an analysis of local bifurcations in the neighborhood of homogeneous equilibrium states is given. For the Dirichlet boundary value problem, conditions under which a pitchfork bifurcation is realized in the neighborhood of the zero equilibrium state are obtained. The more complex nature of bifurcations is realized in the Neumann boundary value problem. It is shown that when the threshold value of the control parameter is exceeded, then from homogeneous equilibrium states, one-parameter families of spatially inhomogeneous equilibrium states are bifurcating. To substantiate the results, the methods of the theory of dynamic systems with an infinite-dimensional space of initial conditions were used: the method of integral (inertial) manifolds, the theory of Poincare normal forms, asymptotic methods of analysis. Their use allows one to obtain asymptotic formulas for the solutions found, and also to study the problem of their stability in the sense of A.M. Lyapunov in the metric of the phase space of solutions.

Keywords:  Cahn-Hilliard equation, boundary value problems, dynamical systems, stability, bifurcations, asymptotic

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