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

Abstract

ABSTRACTWe consider the Cauchy problem 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 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).