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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\f) > 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.

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