A Note on the Asymptotic Behavior of Solutions of the KPP Equation

This note is concerned with the convergence (as $t \to \infty $) to travelling waves of solutions u to the initial value problem of the KPP equation \[u_t = u_{xx} + f(u),\quad x \in \mathbb{R} {\text{ and }}t > 0.\] A travelling wave $\phi _c $ is a solution of the form $u(x,t) = \phi _c (x + ct)$. Estimates for the difference between u and $\phi _c $, in a moving coordinate system $\xi = x + ct$, are given in a weighted supremum norm and in weighted $L^p $-norm $(p \geqq 1)$.