Аннотация
A nonlinear partial differential evolutionary equation is considered, which is obtained as a natural generalization of the well-known Cahn–Hilliard–Oono equation from a physical point of view. The terms responsible for accounting for convection and dissipation have been added to the generalized version. A new version of the equation is considered together with homogeneous Neumann boundary conditions. For such a boundary value problem, local bifurcations of codimension 1 and 2 are studied. In both cases, questions about the existence, stability, and asymptotic representation of spatially inhomogeneous equilibrium states, as well as invariant manifolds formed by such solutions to the boundary value problem, are analyzed. To substantiate the results, the methods of the modern theory of infinite-dimensional dynamical systems, including the method of integral manifolds, the apparatus of the theory of Poincare normal forms, are used. The differences between the results of the analysis of bifurcations in the Neumann boundary value problem are indicated with conclusions in the analysis of the periodic boundary value problem studied by the authors of the article in previous publications.