Approximate Clean Up Lemma

Abstract

In this chapter, we establish some technical results, mainly an “Approximate Clean Up lemma” (valid in arbitrary dimension, which allows us to continue to consider the problems in higher dimensions in the future) and some of its consequences. This lemma, in some sense, describes the little invading property of strongly competing system. For the original Clean Up lemma, see Caffarelli et al. (J. Fixed Point Theory Appl. 5(2), 319–351, 2009). Since these results are only intended for the application to the main result in Chap. 6, and the proof is rather technical, at the first reading the readers need only know the conclusions and directly go to Chap. 6, maybe finally come back to read the details of the proof. We mainly consider the following simplified model.

$$ \frac{\partial u_{i}}{\partial t}-\varDelta u_{i}=-\kappa u_i \sum_{j\neq i} u_j. $$

(5.1)

The original problem (1.1) can be treated with small changes, which we will indicate in Sect. 5.1. This is because (1.1) can be seen as a perturbation of (5.1). The proof of the Approximate Clean Up Lemma follows the iteration scheme used in Caffarelli et al. (J. Fixed Point Theory Appl. 5(2), 319–351, 2009). After establishing this lemma, we also give a linearization version of this lemma (Corollary 5.3.2 and Proposition 5.3.3), by establishing a lower bound for the sum of the two dominating species near the regular part of the free boundaries (Proposition 5.3.1). Finally, we also include a boundary version of the Approximate Clean Up lemma. In this chapter, by saying that a quantity ε(κ) (depending on κ) converges to 0 rapidly, we mean, for some α>0,

$$\varepsilon(\kappa)\leq e^{-\kappa^{\alpha}}. $$