Stationary solutions of non-autonomous Kolmogorov-Petrovsky-Piskunov equations

The paper is devoted to the following problem: w″ (x) + cw′(x) +F (w(x), x) = 0, x ∈ ℝ¹, w(±∞) = w±, where the non-linear term F depends on the space variable x. A classification of non-linearities is given according to the behaviour of the function F (w, x) in a neighbourhood of the points w₊ and w_-. The classical approach used in the Kolmogorov-Petrovsky-Piskunov paper [10] for an autonomous equation (where F = F (u) does not explicitly depend on x), which is based on the geometric analysis on the (w, w′)-plane, is extended and new methods are developed to analyse the existence and uniqueness of solutions in the non-autonomous case. In particular, we study the case where the function F (w, x) does not have limits as x → plusmn;∞.