Convergence to pushed fronts

We study the convergence to the stationary state for the parabolic equation u t = u xx + /(«) with /'(0) > 0 in case there exist front-type solutions U(x + ct) for a continuum of velocities c ^ c(f) and c) > 4/'(0).The initial data are only restricted in their asymptotic behavior for x -* co.We prove strict uniform convergence to a front with velocity c(f) (or a pair of diverging fronts, respectively) with an exponential rate.In [4], Fife and McLeod are concerned with the so-called bistable case, where (/Ì) /'(0) < 0 and /'(l) < 0