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Traveling wave behavior for a nonlinear reaction-diffusion equation. (English) Zbl 1086.34005
Let $\alpha$ and $\beta$ be real nonzero constants; $m$ a positive constant. Applying the bifurcation theory of planar systems, the authors study the bifurcations of bell-profile waves and kink-profile waves for a nonlinear reaction-diffusion equation of the form $$ \frac{ {\partial u}}{{\partial t}}=\alpha \frac{ {\partial^2 u}}{{\partial x}^2}+\beta u (1-u^m). $$ In the last section of the paper, the asymptotic behavior of positive proper solutions for this equation is considered.

34A05Methods of solution of ODE
35B40Asymptotic behavior of solutions of PDE
34C05Location of integral curves, singular points, limit cycles (ODE)
34C23Bifurcation (ODE)
35K57Reaction-diffusion equations