We give a classification of all bounded solutions of the equation [ u'''' + p u'' + F'(u) = 0, qquad -infty < t< infty, ] in which $F$ is a general quartic polynomial and $p$ is restricted to various subsets of $(-infty,0]$. These results are obtained by combining an a priori estimate with geometric arguments in the $(u,u'')$-plane.

Geometric quantization (msc 53D50), Geometry and quantization, symplectic methods (msc 81S10), Geometric methods in differential equations (msc 34A26), Nonlinear equations and systems, general (msc 34A34), General theory (msc 34Axx)
CWI
Modelling, Analysis and Simulation [MAS]

Peletier, M.A. (1999). Non-existence and uniqueness results for the extended Fisher-Kolmogorov equation. Modelling, Analysis and Simulation [MAS]. CWI.