, Volume 75, Issue 4, pp 281-314

A phase plane discussion of convergence to travelling fronts for nonlinear diffusion

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Abstract

The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x,t) of the equation \(u_t - u_{xx} - f\left( u \right) = 0, x \in \left( { - \infty ,\infty } \right),\) in the case f(0)=f(1)=0, with f(u) non-positive for u(>0) sufficiently close to zero and f(u) non-negative for u(<1) sufficiently close to 1. This guarantees the uniqueness (but not the existence) of a travelling front solution u;U(x−ct), U(−∞);0, U(∞);, and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front, if it exists, and to a “stacked” combination of travelling fronts if it does not. The approach is to use the monotonicity to take u and t as independent variables and p = u x as the dependent variable, and to apply ideas of sub- and super-solutions to the diffusion equation for p.

This research was sponsored by the United States Army under Contract No. DAAG29-75-C-0024.