Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on RN

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:262

Article

Salako, Rachidi Bolaji¹  Shen, Wenxian¹ 

[1] Auburn Univ, Dept Math & Stat, Auburn, AL 36849 USA

关键词: Parabolic elliptic chemotaxis system;    Logistic source;    Classical solution;    Local existence;    Global existence;    Asymptotic behavior;   

DOI  :  10.1016/j.jde.2017.02.011

来源: Elsevier

【 摘 要 】

In the current paper, we consider the following parabolic elliptic semilinear Keller Segel model on R-N, {u(t) = del center dot(del(u) - chi(u)del(u)) + a(u) - b(u)(2), x is an element of R-N, t > 0, 0=(Delta - I)v + u, x is an element of R-N, t > 0, where x > 0, a >= 0, b > 0 are constant real numbers and N is a positive integer. We first prove the local existence and uniqueness of classical solutions (u(x, t; u(0)), v(x, t; u(0))) with u(x, 0; u(0)) = u(0) (x) for various initial functions u(0) (x). Next, under some conditions on the constants a, b, x and the dimension N, we prove the global existence and boundedness of classical solutions (u(x, t; u(0)), v(x, t; u(0))) for given initial functions u(0)(x). Finally, we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions. Under some conditions on the constants a, b, x and the dimension N, we show that for every strictly positive initial function u(0)(.), lim(t ->infinity x is an element of R)(N) sup [vertical bar u(x,y;u(0)) - a/b vertical bar + vertical bar v(x,t;u(0)) - a/b vertical bar = 0, and that for every nonnegative initial function u0(.) with non -empty and compact support supp(u0), there are 0 < c(low)(*)row (u(0)) < c(up)(*)(u(0)) < infinity such that lim(t ->infinity vertical bar x vertical bar <= ct) sup [vertical bar u(x, t ;u(0)) - a/b vertical bar + vertical bar v(x, t; u(0)) - a/b vertical bar = 0 for all(0) < c < c(low)(*)(u(0)) and lim(t ->infinity vertical bar x vertical bar >= ct) sup [vertical bar u(x, t; u(0)) - a/b vertical bar + vertical bar v(x, t; u(0)) - a/b vertical bar = 0 for all(c) > c(up)(*)(u(0)). (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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