Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities

We consider the Cauchy problem (Formula presented.) where f is a C¹ function on (Formula presented.) with (Formula presented.) and u₀ is a nonnegative continuous function on (Formula presented.) whose limits at (Formula presented.) are equal to 0. Assuming that the solution u is bounded, we study its asymptotic behavior as (Formula presented.) In the first part of this study, we proved a general quasiconvergence result: as (Formula presented.) the solution approaches a set of steady states in the topology of (Formula presented.) In this paper, we show that under certain generic, explicitly formulated conditions on the nonlinearity f, the solution necessarily converges to a single steady state (Formula presented.) in (Formula presented.) Then, under the same conditions, we describe the global asymptotic shape of the solution: the graph of (Formula presented.) has a top part close to the graph of (Formula presented.) and two sides taking shapes of “terraces” moving in the opposite directions with precisely determined speeds.