Existence and qualitative properties of entire solutions of the KPP equation. (Existence et propriétés qualitatives de solutions entières de l’équation de KPP.) (French. Abridged English version) Zbl 0913.35143
Summary: This note deals with the solutions defined for all time (i.e. entire) of
\[
u_t= u_{xx}+ f(u),\quad 0< u(x,t)<1,\quad x\in\mathbb{R},\quad t\in\mathbb{R},
\]
where \(f\) is a KPP type nonlinearity on \([0,1]\). This equation admits infinitely many travelling waves type solutions as well as solutions of the type \(u(t)\). We have build four other manifolds of solutions, the biggest one being five-dimensional. Furthermore, the travelling waves are on the boundary of these four manifolds. We also answer the question of the uniqueness for a certain class of solutions.
MSC:
35Q80 | Applications of PDE in areas other than physics (MSC2000) |
35K57 | Reaction-diffusion equations |