Abstract:

It this paper,author studies the traveiling-wave solutions of the reaction-diffusion e-quation
u_t=u_xx+u(1-u)(u-a)(1)where 0 < a < 1,and c ≥ 0.Four types of traveiling-wave solutions are obtained with theirminimal or unique wave speeds c. Explicit form of some wave-front solutions including theHuxley wave are also established. The results are then applied to investigate the following pla-nar cubic system ζ=η, η=aξ+cη+(1+a)ξ²+ζ³.(2)It is well known that in the (U,U') phaes plane,the equation (1)is equivalent to the system(2). H. P. Mckean studied (1) by means of the traveiling-wave solution tnethod,and acquiredsome phase portraits for a∈(0,1/2) and c ≥ 0.Their results in this paper,however,glve allthe phase portraits for system (2) together with the bifurcation diagram.

Key words: Reaction-diffusion equation, traveiling-wave solution, explicit wave-front solution, wave sneed, planar nolynomlalsvstem, global phase portrait, bifurcation diagram