Propagation or extinction in bistable equations: the non-monotone role of initial fragmentation

Abstract

In this paper, we investigate the large-time behavior of bounded solutions of the Cauchy problem for a reaction-diffusion equation in ${\mathbb{R}}^N$ with bistable reaction term. We consider initial conditions that are chiefly indicator functions of bounded Borel sets. We examine how geometric transformations of the supports of these initial conditions affect the propagation or extinction of the solutions at large time. We also consider two fragmentation indices defined in the set of bounded Borel sets and we establish some propagation or extinction results when the initial supports are weakly or highly fragmented. Lastly, we show that the large-time dynamics of the solutions is not monotone with respect to the considered fragmentation indices, even for equimeasurable sets.

References in corpus (4)

• Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations
• Quantitative estimates of the threshold phenomena for propagation in reaction-diffusion equations
• Radial terrace solutions and propagation profile of multistable reaction-diffusion equations over ${\mathbb{R}}^N$
• Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations

Cited by in corpus (2)

• On the instability of threshold solutions of reaction-diffusion equations, and applications to optimization problems
• Long time behavior of the field-road diffusion model: an entropy method and a finite volume scheme