We consider several problems for degenerate parabolic equations exhibiting nonlinearities of various kinds. For example the equations may contain superlinear sources, causing blow up of the solutions, or damping terms; the principal part of the operator is also nonlinear. We mention as unifying features the fact that the spatial domains have non-compact boundary, and the technical approach which is based on energy methods and a priori estimates. The issues investigated include existence under optimal assumptions on the data, asymptotic behavior of solutions, existence or non-existence of global in time solutions.

In this paper, we study a phenomenological model for pattern formation in electroconvection, and the effect of noise on the pattern. As such model we consider an anistropic Swift–Hohenberg equation adding an additive noise. We prove the existence of a global solution of that equation on the two dimensional torus. In addition, inserting a scaling parameter, we consider the equation on a large domain near its change of stability. We observe numerically that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a stochastic Ginzburg–Landau equation.