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We study the Cauchy problem where f (u) is a locally Lipschitz continuous function satisfying f (0) = 0. We show that any nonnegative bounded solution with compactly supported initial data converges to a stationary solution as t → ∞. Moreover, the limit is either a constant or a symmetrically decreasing stationary solution. We also consider the special case where f is a bistable nonlinearity and the case where f is a combustion type nonlinearity. Examining the behavior of a parameter-dependentdoi:10.4171/jems/198 fatcat:6yy7m7uqjrclvn5rgvebny7xna