Approximate Clean Up Lemma
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,
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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. ∂ui uj . − Δui = −κui ∂t
(5.1)
j =i
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, ε(κ) ≤ e−κ . α
5.1 Systems with Zeroth Order Terms In the paper [2], a Clean Up lemma was established (see their Sect. 3, Theorem 11) for the following system on a domain in Rn : K. Wang, Free Boundary Problems and Asymptotic Behavior of Singularly Perturbed Partial Differential Equations, Springer Theses, DOI 10.1007/978-3-642-33696-6_5, © Springer-Verlag Berlin Heidelberg 2013
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Approximate Clean Up Lemma
⎧ Δui ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Δ ui − uj ≤ 0, (5.2)
j =i
⎪ ⎪ ⎪ ⎪ ui ≥ 0, ⎪ ⎪ ⎪ ⎩ ui uj = 0,
for i = j.
We need to consider a system with zeroth order perturbation. Assume we are in the unit ball B1 (0) ⊂ Rn . ⎧ −Δui ≤ fi (ui ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−Δ ui − uj ≥ fi (ui ) − fj (uj ), (5.3) j =i j =i ⎪ ⎪ ⎪ ⎪ui ≥ 0, ⎪ ⎪ ⎪ ⎩ ui uj = 0, for i = j. Here fi , 1 ≤ i ≤ M, are given Lipschitz continuous function defined on R+ with fi (u) = ai u − u2 (we can allow more general nonlinearity, which we do not pursue here). Then we have the following “Clean Up Lemma”. Theorem 5.1.1 Assume at 0 ∈ B1 (0), there exists a sequence λk → 0 such that the vector function λ1k u(λk x) converges to uˆ 1 = αx1+ , Then in a neighborhood of 0
uˆ 2 = αx1− ,
uˆ j = 0
for j > 2.
uj ≡ 0.
j >2
Remark 5.1.2 It is easy to see that the above blow up limit satisfies the system (5.2) on Rn (using the fact fi (0) = 0). The proof of this theorem is almost the same to the one in [2], because, after restricting to a small ball and rescaling
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