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

Abstract. We consider the Cauchy problem We consider the Cauchy problem ut = uxx + f(u), x ∈ R, t > 0, u(x, 0) = u0(x), x ∈ R, 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 ∗Supported in part by NSF Grant DMS–1161923