The correspondence between travelling-wave solutions of a nonlinear reaction-convection-diffusion equation and an integral equation. (English) Zbl 0855.35061

Gilding, B. H.

Differ. Integral Equ. 9, No. 5, 919-947 (1996).

The author examines the nonlinear reaction-convection-diffusion equation \[ u_t= (a(u))_{xx}+ (b(u))_x+ c(u). \] Under mild continuity conditions on the coefficients \(a\), \(b\) and \(c\), it is shown that the existence of travelling wave solutions, of the form \(f(x- \lambda t)\) (for suitably restricted \(f\)), is equivalent to the existence of a solution \(\theta\) of the singular nonlinear integral equation \[ \theta(s)= \lambda s+ b(s)- \int^s_0 {c(r)\over \theta(r)} da(r). \] A number of consequences of this correspondence between the integral equation and travelling waves are discussed. In particular, an open question in the theory of finite speed of propagation for the reaction-convection-diffusion equation is resolved.

Reviewer: W.Lamb (Glasgow)

MSC:

35K55	Nonlinear parabolic equations
35K57	Reaction-diffusion equations

Keywords:

nonlinear reaction-convection-diffusion equation; travelling wave solutions