Global boundedness of solutions to a two-species chemotaxis system

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 66; no. 1; pp. 83 - 93
• Main Authors: Zhang, Qingshan, Li, Yuxiang
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.02.2015
• Subjects: Boundary conditions; Engineering; Functions (mathematics); Logistics; Mathematical analysis; Mathematical Methods in Physics; Theoretical and Applied Mechanics; Two-species chemotaxis system; 35B35; Boundedness; Logistic source; Global existence; 92C17; 35K55
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-013-0383-4

In this paper, we consider the chemotaxis system of two species which are attracted by the same signal substance u t = Δ u - ∇ · ( u χ 1 ( w ) ∇ w ) + μ 1 u ( 1 - u - a 1 v ) , x ∈ Ω , t > 0 , v t = Δ v - ∇ · ( v χ 2 ( w ) ∇ w ) + μ 2 v ( 1 - a 2 u - v ) , x ∈ Ω , t > 0 , w t = Δ w - w + u + v , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n . We prove that if the nonnegative initial data ( u 0 , v 0 ) ∈ ( C 0 ( Ω ¯ ) ) 2 and w 0 ∈ W 1 , r ( Ω ) for some r >  n , the system possesses a unique global uniformly bounded solution under some conditions on the chemotaxis sensitivity functions χ 1 ( w ),  χ 2 ( w ) and the logistic growth coefficients  μ 1 ,  μ 2 .