Dynamics of nonnegative solutions of one-dimensional reaction–diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences

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).