Bifurcations of spatially inhomogeneous solutions in a modified version of the Kuramoto–Sivashinsky equation

A periodic boundary-value problem for an equation with a deviating spatial argument is considered. Using the Poincaré–Dulac method of normal forms, the method of integral manifolds, and asymptotic formulas, we examine a number of bifurcation problems of codimension $1$ and $2$. For homogeneous equilibrium states, we analyze possibilities of realizing critical cases of various types. The problem on the stability of homogeneous equilibrium states is studied and asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained.