Global Travelling Waves in Reaction–Convection–Diffusion Equations

We study the existence and properties of solutions in travelling wave form, u(x, t)=φ(x−st), defined for every z=x−st , for the reaction-convection- diffusion equation with a, m, n>0; b, k, p . In the reaction case k>0 we prove that there exist travelling waves vanishing for z→∞ if and only if b>0 and Moreover, if m+p≠2n, there exists a minimal velocity s*(a, b, k, m, n, p)>0, for which there are travelling waves only with s s*, while in the case m+p=2n there are travelling waves only when 4amk b2n and for every velocity s 0. Some properties of the function s* are established. All the waves are decreasing in their support and waves having bounded support from the right exist if and only if m>min{1, n}. Also, the absorption case k<0 is treated, where we find that, for different values of the parameters, there exists a unique travelling wave for every velocity s , but for some case where only negative velocities exist. The cases b=0 or k=0 are well known in the literature.