Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system

Abstract

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every u 0 ∈ L 1 (R 2 ). The local existence time is characterized for u 0 ∈ L 1 ∩ L q * (R 2 ) with 1 < q * < 2. Next, we prove the finite time blow-up of strong solution under the assumption u 0 L 1 > 8π and x| 2 u 0 L 1 < 1 γ ·g( u 0 L 1 /8π), where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.

Diffusion-dominated asymptotics of solution to chemotaxis model

“…Although the mainstream of research for the whole plane case is devoted to the problem of finite time blowup, the elements of global existence, under smallness assumption on initial mass, can be found in [11,21], and [18] A related result is obtained in the paper [18], where general type of the aggregation equation, i.e., of the form 4) for various kernels K is considered. The existence results are given for two kinds of kernels: mildly singular (such that ∇ K ∈ L q (R n ) for some q ∈ (n, ∞]) and strongly singular (if ∇ K ∈ L q (R n ) for some q ∈ (1, n] and ∇ K ∈ L p (R n ) for every p > n).…”

“…Since the kernel K (given by (1.2)) is an example of a strongly singular kernel, under smallness assumption on u 0 1 , Theorem 2.6 in [18] implying global existence and decay as in [21] can be applied.…”

“…As far as we know, there is no result (except that in [21] and [18] contained in the existence results) concerning the asymptotic behavior of the solution to the considered problem. The problem of the asymptotic profile for the problem with parabolic version of (1.2) was studied in the papers [20,25] and [26], under the general assumption that L 1 and L ∞ norms of solutions are bounded.…”

“…Using this dependence, we modify Theorem 3.1 in a way presented in [21] to obtain existence and uniqueness of the solutions, under smallness condition on u 0 1 . Granted the result on the time decay included in the definition of X p , we establish required time decay of the solutions for all indices p.…”

Existência de solução para a equação de agregação com dado inicial em espaços de Morrey

“…Remetemos o leitoràs obras [3,4,8], e as referências nelas contidas, para resultados matemáticos da modelagem de sistemas de quimiotaxia. Vamos considerar o problema de Cauchy (1)-(2) e mostrar a existência de solução global no tempo, com a condição inicial em espaços de Morrey, no caso em que n ≥ 2 e que a função Ké tal que ∇K ∈ L 1 (R n ).…”