Global travelling waves in reaction-convection-diffusion equations

We study the existence and properties of solutions in travelling wave form, u(x, t) = phi(x - st), defined for every z = x - st is an element of R, for the reaction-convection-diffusion equation u(t) = a(u(m))(xx) + b(u(n))(x) + ku(p) for x is an element of R, t > 0, with a, m, n > 0; b, k, p is an element of R. In the reaction case k > 0 we prove that there exist travelling waves vanishing for z --> infinity if and only if b > 0 and min {1, n} less than or equal to m + p/2 less than or equal to maxi {1, n}, p less than or equal to max {1, n}. Moreover, if m + p not equal 2n, there exists a minimal velocity s*(a, b, k, m, n, p) > 0, for which there are travelling waves only with s greater than or equal to s*, while in the case m + p = 2n there are travelling waves only when 4amk less than or equal to b(2)n and for every velocity s greater than or equal to 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 is an element of R, but for some case where only negative velocities exist. The cases b = 0 or k = 0 are well known in the literature. (C) 2000 Academic Press.