The correspondence between travelling-wave solutions of a nonlinear reaction-convection-diffusion equation and an integral equation
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.
nonlinear reaction-convection-diffusion equation
scientific article; zbMATH DE number 896467