The approach of solutions of nonlinear diffusion equations to travelling front solutions

The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation u_t—u_xx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given.

1. Let u(x, 0)=u ₀(x) satisfy 0≦u ₀≦1. Let (a_ = mathop {lim sup u0}limits_{x to - infty } {text{(}}x{text{), }}mathop {lim inf u0}limits_{x to infty } {text{(}}x{text{)}}) . Then u approaches a translate of U uniformly in x and exponentially in time, if a_? is not too far from 0, and a₊ not too far from 1.
2. Suppose (intlimits_{text{0}}^{text{1}} {f{text{(}}u{text{)}}du} > {text{0}}) . If a _? and a ₊ are not too far from 0, but u₀ exceeds a certain threshold level for a sufficiently large x-interval, then u approaches a pair of diverging travelling fronts.
3. Under certain circumstances, u approaches a “stacked” combination of wave fronts, with differing ranges.