Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 65; no. 6; pp. 1137 - 1152
• Main Authors: Wang, Liangchen, Mu, Chunlai, Zhou, Shouming
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.12.2014
• Subjects: Bacteria; Boundary conditions; Boundary value problems; Concentration gradient; Diffusion; Engineering; Mathematical analysis; Mathematical Methods in Physics; Mathematical models; Nonlinearity; Theoretical and Applied Mechanics; Global existence; 92C17; 35B35; Chemotaxis; 35K55; Boundedness
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-013-0375-4

This paper deals with the global existence and boundedness of the solutions for the quasilinear chemotaxis system u t = ∇ · ( D ( u ) ∇ u ) - ∇ · ( u χ ( v ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v - u f ( v ) , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω ⊂ R n , with nonnegative initial data u 0 ∈ W 1 , θ ( Ω ) (for some θ >  n ) and v 0 ∈ W 1 , ∞ ( Ω ) . The given functions D ( s ) , χ ( s ) and f ( s ) are supposed to be sufficiently smooth for all s  ≥ 0 and such that f (0) = 0. This model describes the motion of the cells (e.g., bacteria) under the effect of gradients of the concentration of the oxygen that is consumed by the cells. It is proved that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in Ω × ( 0 , + ∞ ) provided that some technical conditions are fulfilled.