We consider the Cauchy problem (Formula presented.) where f is a locally Lipschitz function on ℝ with f(0) = 0, and u0 is a nonnegative function in C0(ℝ), the space of continuous functions with limits at ± ∞ equal to 0. Assuming that the solution u is bounded, we study its large-time behavior from several points of view. One of our main results is a general quasiconvergence theorem saying that all limit profiles of u(·, t) in (Formula presented.) are steady states. We also prove convergence results under additional conditions on u0. In the bistable case, we characterize the solutions on the threshold between decay to zero and propagation to a positive steady state and show that the threshold is sharp for each increasing family of initial data in C0(ℝ).
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- generalized omega-limit set
- localized initial data
- parabolic equations on ℝ
- threshold solutions