Periodic problem for an evolution equation with a quadratic and a cubic nonlinearity

We study conditions for the existence of a solution of a periodic problem for a model nonlinear equation in the spatially multidimensional case and consider various types of large time asymptotics (exponential and oscillating) for such solutions. The generalized Kolmogorov-Petrovskii-Piskunov equation, the nonlinear Schrödinger equation, and some other partial differential equations are special cases of this equation. We analyze the solution smoothing phenomenon under certain conditions on the linear part of the equation and study the case of nonsmall initial data for a nonlinearity of special form. The leading asymptotic term is presented, and the remainder in the asymptotics of the solution is estimated in a spatially uniform metric.