Diffusive waves in inhomogeneous media

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Abstract
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 equationIn addition, there are travelling wave solutions u+(x, t) and u̲(x, t) which, ifconnect h+ to h̲ in the sense thatthe convergence being uniform on bounded x-intervals. These solutions are of the formwhere 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.

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