TY - JOUR
T1 - Stationary solutions of non-autonomous Kolmogorov-Petrovsky-Piskunov equations
AU - Volpert, V. A.
AU - Soukhov, Iouri M.
PY - 1999/1/1
Y1 - 1999/1/1
N2 - 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;∞.
AB - 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;∞.
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U2 - 10.1017/S0143385799138823
DO - 10.1017/S0143385799138823
M3 - Article
AN - SCOPUS:0033417246
VL - 19
SP - 809
EP - 835
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
SN - 0143-3857
IS - 3
ER -