A new result for global solvability and boundedness in the N-dimensional quasilinear chemotaxis model with logistic for source and consumption of chemoattractant

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:475

Article

Song, Xinchao¹  Zheng, Jiashan^1,2 

[1] Ludong Univ, Sch Math & Stat Sci, Yantai 264025, Peoples R China

[2] Renmin Univ China, Sch Informat, Beijing 100872, Peoples R China

关键词: Boundedness;    Chemotaxis;    Global existence;    Logistic source;    Consumption of chemoattractant;   

DOI  :  10.1016/j.jmaa.2019.03.002

来源: Elsevier

【 摘 要 】

We consider the following chemotaxis model {u(t) = del . (D(u)del u) - chi del . (u del v) +mu(u - u(2)), x is an element of Omega, t > 0, v(t) - Delta v = -uv, x is an element of Omega, t > 0, (D(u)del u - chi u . del v) . nu = partial derivative v/partial derivative nu = 0, x is an element of partial derivative Omega, t > 0, u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), x is an element of Omega on a bounded domain Omega subset of R-N (N >= 1), with smooth boundary partial derivative Omega, chi and mu are positive constants. Here, D is supposed to be smooth positive function satisfying D(u) >= C-D(u + 1)(m-1) for all u >= 0 with some C-D, m > 0. Besides appropriate smoothness assumptions, in this paper it is only required that m >{1 - mu/2 chi[1+max(1 <= 0 <= 2) lambda(0)(s) parallel to v(0 )parallel to (L)infinity(Omega) 2(3)]( )if( )N <= 2(, ) 1 if N >= 3, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where lambda(0) is a positive constant which is corresponding to the maximal Sobolev regularity. The results of this paper extends the results of Jin (J. Differential Equations 263 (9) (2017) 5759-5772), who proved the possibility of boundness of weak solutions, in the case m > 1 and N = 3. To the best of our knowledge, this is the first result which gives the relationship between m and mu/x that yields to the boundedness of the solution. (C) 2019 Elsevier Inc. All rights reserved.

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