Boundedness in a quasilinear chemotaxis–haptotaxis model of parabolic–parabolic–ODE type

• Published in: Boundary value problems Vol. 2019; no. 1; pp. 1 - 18
• Main Authors: Lei, Long, Li, Zhongping
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 27.08.2019; Hindawi Limited; SpringerOpen
• Subjects: Analysis; Approximations and Expansions; Boundary conditions; Boundedness; Chemotaxis; Difference and Functional Equations; Haptotaxis; Logistic source; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinear diffusion; Nonlinear production; Ordinary Differential Equations; Parameters; Partial Differential Equations; Haptotaxis; Boundedness; Logistic source; 35B65; 92C17; Nonlinear diffusion; Chemotaxis; Nonlinear production; 35K55; 35Q92
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-019-1255-4

This paper deals with the boundedness of solutions to the following quasilinear chemotaxis–haptotaxis model of parabolic–parabolic–ODE type: { u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u r − 1 − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u η , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 , under zero-flux boundary conditions in a smooth bounded domain Ω ⊂ R n ( n ≥ 2 ) , with parameters r ≥ 2 , η ∈ ( 0 , 1 ] and the parameters χ > 0 , ξ > 0 , μ > 0 . The diffusivity D ( u ) is assumed to satisfy D ( u ) ≥ δ u − α , D ( 0 ) > 0 for all u > 0 with some α ∈ R and δ > 0 . It is proved that if α < n + 2 − 2 n η 2 + n , then, for sufficiently smooth initial data ( u 0 , v 0 , w 0 ) , the corresponding initial-boundary problem possesses a unique global-in-time classical solution which is uniformly bounded.