When the function f(u) is of “bistable type’, i.e. has two zeros h̲ and h+ at which f' is negative and (for simplicity) has only one other zero between them, then the constant functions u = h± are L∞-stable solutions of the nonlinear diffusion equation
In addition, there are travelling wave solutions u+(x, t) and u̲(x, t) which, if
connect h+ to h̲ in the sense that
the convergence being uniform on bounded x-intervals. These solutions are of the form
where U(z) is a monotone function (the wave's profile), U(±∞) = h±, and the velocity c is a specific positive number depending on the function f.