Global smooth solvability of a parabolic-elliptic nutrient taxis system in domains of arbitrary dimension

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Tao, Youshan¹  Winkler, Michael² 

[1] Donghua Univ, Dept Appl Math, Shanghai 200051, Peoples R China

[2] Univ Paderborn, Inst Math, D-33098 Paderborn, Germany

关键词: Nutrient taxis;    Prey taxis;    Global smooth solution;    Boundedness;    Stabilization;   

DOI  :  10.1016/j.jde.2019.01.014

来源: Elsevier

【 摘 要 】

This paper deals with the nutrient taxis system u(t) = Delta u - del . (u del v), 0 = Delta v - uv - mu v + r( x, t), in a bounded domain Omega subset of R-n, n >= 1, with smooth boundary, where mu >= 0 is a parameter and r is an element of C-1((Omega) over barx[ 0, infinity)) is a given nonnegative function. It is shown that for any prescribed initial data u(0) is an element of W-1,W-infinity (Omega) with u(0) > 0 in (Omega) over bar, the corresponding Neumann initial-boundary problem admits a global classical solution. With regard to qualitative aspects, it is moreover, inter alia, seen that if radditionally satisfies integral(t+1)(t) integral(Omega) vertical bar del root r vertical bar(2) -> 0 as t -> infinity, then in the large time limit the solution component ustabilizes toward the constant 1/vertical bar Omega vertical bar integral(Omega)u(0) with respect to the norm in L-1(Omega), and that if furthermore supt>0 parallel to r(.,t)parallel to(Lq) (Omega) < infinity for some q >= 1 fulfilling q > n/2, then u is uniformly bounded. (c) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free