Convergence to travelling fronts in semilinear parabolic equations

Synopsis We study the convergence to the stationary state for the parabolic equation u , = u xx + F(u) . There exist wave-type solutions u(x, t) = φ( x − ct ) for a continuum of velocities c . In the asymptotic behavior of this equation was investigated for a step function as initial data. In this paper we obtain the asymptotic behavior for a large class of monotone initial data. All solutions with initial data in this class evolve to wave-type solutions, where the rate of decay of the initial data determines the asymptotic speed.