Thresholds, fragmentation and symmetrization in parabolic equations

Abstract

This paper is mainly concerned with the large-time dynamics of bounded nonnegative solutions of reaction-diffusion equations on the real line with nonlinearities mainly of the bistable-type. We first consider initial data of the type $\alpha \mathbf{1}_I$, namely scalar multiples of indicator functions of intervals $I$. For each amplitude $\alpha$, the existence of a threshold length $L^*(\alpha)$ separating the extinction and the persistence of the solutions at large time is known. We here address the question as to whether the limit of the threshold sizes $L^*(\alpha)$ as $\alpha\to+\infty$ is positive or zero. We provide sufficient conditions under which this limit is positive, and others under which it is zero. Secondly, when the limit is positive, we show that some fragmented initial data, which are equally distributed as $\alpha \mathbf{1}_I$, give rise to solutions persisting at large time, whereas the solutions emanating from $\alpha \mathbf{1}_I$ go to extinction at large time. This result shows an unexpected favourable effect of the fragmentation of the initial datum on the large-time dynamics. Last, we consider the mass concentration principle for parabolic equations, which states that nonnegative solutions can be controlled from above in an integral sense by the solutions emanating from the Schwarz symmetrically decreasing rearrangements of the initial data. Since the pioneering results of [Bandle, 1976] and [Alvino, Trombetti, Lions, 1990], this principle is known to hold in bounded domains with Dirichlet boundary conditions under different types of assumptions of the coefficients of the equation. We show that this mass concentration principle is not valid in general, even for simple equations of the type $\partial_t u = \partial_{xx} u + f(u)$.