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In this chapter we discuss the blowup mechanism of the nonstationary simplified system of chemotaxis. This chapter is devoted to the proof of Theorem 1.1.

In this chapter we show that the mass quantization of collapse occurs if the solution [148]. It is uncertain whether such a solution exists or not. Actually, it is suspected that the blowup in infinite time occurs only when the solution converges to a singular stationary solution, and therefore, in that case, the total mass λ = ||O|| must be quantized as λ ∈ 4N. This question is open, but an important tool is exploited, which we call the (This is different from the lemma given by [23].)

Our concern is focused on the problem of mass quantization, which arises as . Here, u = u(x, t) denotes the classical solution to (11.2): where Ω ⊂ R is a bounded domain with smooth boundary ∂Ω, and a > 0 is a constant.

We are concerned with the classical solution u = u(x, t) to (11.2): and study the problem of mass quantization, in (11.3): as t ↑ T. Here, Ω ⊂ R is a bounded domain with smooth boundary ∂Ω, a > 0 is a constant, and

Motivated by Theorem 14.1, we introduced the standard backward self-similar transformation in the previous chapter. Supposing T = T < +∞ and x ∈ S, we define R(t) = (T-t)1/2, y = (x-x)/R(t), and s = –log(T-t). Then z(y, s) = (T – t)u(x, t) satisfies (14.11), which is a similar system to (14.1), and this {z(·, s)} is regarded as its global semiorbit. Similarly to the collapse formed in infinite time in the prescaled system, quantized are formed in infinite time in this rescaled system, stated as Theorem 14.2. Thus, any s →+∞admits a subsequence {s′} ⊂ {s} satisfying (14.12) in M (R): where supp µ(dy) , , and

This chapter is the epilogue. We summarize the argument and give a new formulation applicable to other theories.