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Boundedness in a quasilinear chemotaxis‐haptotaxis system with logistic source
Author(s) -
Wang Liangchen,
Mu Chunlai,
Hu Xuegang,
Tian Ya
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4216
Subject(s) - bounded function , mathematics , domain (mathematical analysis) , neumann boundary condition , homogeneous , degenerate energy levels , diffusion , boundary (topology) , pure mathematics , mathematical analysis , combinatorics , physics , thermodynamics , quantum mechanics
This paper studies the chemotaxis‐haptotaxis system with nonlinear diffusionu t = ∇ · ( D ( u ) ∇ u ) − χ ∇ · ( u ∇ v ) − ξ ∇ · ( u ∇ w ) + μ u ( 1 − u − w ) ,x ∈ Ω ,t > 0 ,0 = Δ v − v + u ,x ∈ Ω ,t > 0 ,w t = − v w ,x ∈ Ω ,t > 0 ,subject to the homogeneous Neumann boundary conditions and suitable initial conditions, where χ , ξ and μ are positive constants, and Ω ⊂ R n ( n ⩾2) is a bounded and smooth domain. Here, we assume that D ( u )⩾ c D u m − 1 for all u > 0 with some c D > 0 and m ⩾1. For the case of non‐degenerate diffusion, if μ > μ ∗ , whereμ ∗ : =( 2 − m ) n − 2 ( 2 − m ) n χ ,if m < 2 − 2 n ,0 ,if m ⩾ 2 − 2 n ,it is proved that the system possesses global classical solutions which are uniformly‐in‐time bounded. In the case of degenerate diffusion, we show that the system admits a global bounded weak solution under the same assumptions. Copyright © 2016 John Wiley & Sons, Ltd.