Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on RN. II. Existence, uniqueness, and stability of strictly positive entire solutions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:464

Article

Salako, Rachidi B.¹  Shen, Wenxian¹ 

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

关键词: Parabolic-elliptic chemotaxis system;    Logistic source;    Stability;    Entire solutions;    Asymptotic spreading;    Comparison principle;   

DOI  :  10.1016/j.jmaa.2018.04.034

来源: Elsevier

【 摘 要 】

The current work is the second of the series of three papers devoted to the study of asymptotic dynamics in the following parabolic-elliptic chemotaxis system with space and time dependent logistic source, {partial derivative(t)u = Delta u - chi del center dot (u del v) + u(a(x,t) - ub(x,t)), x is an element of R-N, 0 = Delta v - lambda v + mu u, x is an element of R-N, (0.1) where N >= 1 is a positive integer, chi, lambda and mu are positive constants, and the functions a(x, t) and b(x, t) are positive and bounded. In the first of the series, we studied the phenomena of pointwise and uniform persistence, and asymptotic spreading in (0.1) for solutions with compactly supported or front like initials. In the second of the series, we investigate the existence, uniqueness and stability of strictly positive entire solutions of (0.1). In this direction, we prove that, if 0 <= mu chi < inf(x,t) b(x, t), then (0.1) has a strictly positive entire solution, which is time-periodic (respectively time homogeneous) when the logistic source function is time-periodic (respectively time homogeneous). Next, we show that there is positive constant chi(0), depending on N, lambda, mu, a and b such that for every 0 <= chi < chi(0), (0.1) has a unique positive entire solution which is uniform and exponentially stable with respect to strictly positive perturbations. In particular, we prove that chi(0) can be taken to be inf(x,t) b(x,t)/2 mu when the logistic source function is either space homogeneous or the function (x, t) bar right b(x,t)/a(x,t) is constant. We also investigate the disturbances to Fisher-KKP dynamics a(x,t) caused by chemotactic effects, and prove that sup (0= 0) 1/chi vertical bar vertical bar u(chi) (center dot, t + t(0); t(0), u(0)) - u(0)(center dot, t + t(0); t(0), u(0)) vertical bar vertical bar(infinity) 0, where (u(chi)(x, t t(0); t(0), n(0)), v(chi)(x, t t(0); t(0), u(0))) denotes the unique classical solution of (0.1) with u(chi)(x,t(0); t(0), u(0)) = u(0)(x), for every 0 <= chi < b(inf). (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free