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In this paper, we consider the quasilinear chemotaxis–haptotaxis system $$\begin{aligned}\left\{\begin{array}{ll}u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(S_1(u)\nabla v)-\nabla\cdot(S_2(u)\nabla w)+uf(u,w),\quad x\in\Omega,~t > 0,v_t=\Delta v-v+u,\quad x\in\Omega,~t > 0,w_t=-vw,\quad x\in\Omega,~t > 0\end{array} \right.\end{aligned}$$ (⋆)in a bounded smooth domain \({\Omega\subset\mathbb{R}^n~(n\geq1)}\) under zero-flux boundary conditions, where the nonlinearities \({D,~S_1}\) and \({S_2}\) are assumed to generalize the prototypes $$D(u)=C_{D}(u+1)^{m-1},~S_1(u)=C_{S_1}u(u+1)^{q_1-1} \quad {\mathrm{and}} \quad S_2(u)=C_{S_2}u(u+1)^{q_2-1}$$with \({C_D,C_{S_1},C_{S_2} > 0,~m,q_1,q_2\in\mathbb{R}}\) and \({f(u,w)\in C^1([0,+\infty)\times[0,+\infty))}\) fulfills $$f(u,w)\leq r-bu\quad {\mathrm{for all}}~~u\geq 0\quad {\mathrm{and}} \quad w\geq 0,$$where \({r > 0,~b > 0.}\) Assuming nonnegative initial data \({u_0(x)\in W^{1,\infty}(\Omega),v_0(x)\in W^{1,\infty}(\Omega)}\) and \({w_0(x)\in C^{2,\alpha}(\bar\Omega)}\) for some \({\alpha\in(0,1),}\) we prove that (i) for \({n\leq2,}\) if \({\max\{q_1,q_2\} < m+\frac{2}{n}-1,}\) then \({(\star)}\) has a unique nonnegative classical solution which is globally bounded, (ii) for \({n > 2,}\) if \({\max\{q_1,q_2\} < m+\frac{2}{n}-1}\) and \({m > 2-\frac{2}{n}}\) or \({\max\{q_1,q_2\} < m+\frac{2}{n}-1}\) and \({m\leq 1,}\) then \({(\star)}\) has a unique nonnegative classical solution which is globally bounded. Keywords Chemotaxis Haptotaxis Quasilinear Boundedness Logistic source Mathematics Subject Classification 35B65 35K55 35Q92 92C17 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (45) References1.Alikakos N.D.: Bounds of solutions of reaction-diffusion equations. Commun. Partial Differ. Eqn. 4, 827–868 (1979)MathSciNetCrossRefMATH2.Biler P.: Local and global solvability of some parabolic systems modelling chemotaxis. Adv. Math. Sci. Appl. 8, 715–743 (1998)MathSciNetMATH3.Calvez V., Carrillo J.A.: Volume effects in the Keller-Segel model: energy estimates preventing blow-up. J. Math. 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TMA 72, 1044–1064 (2010)MathSciNetCrossRefMATH45.Wrzosek D.: Global attractor for a chemotaxis model with prevention of overcrowding. Nonlinear Anal. TMA 59, 1293–1310 (2004)MathSciNetCrossRefMATH About this Article Title Boundedness in a quasilinear chemotaxis–haptotaxis system with logistic source Journal Zeitschrift für angewandte Mathematik und Physik 67:21 Online DateApril 2016 DOI 10.1007/s00033-016-0620-8 Print ISSN 0044-2275 Online ISSN 1420-9039 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Theoretical and Applied Mechanics Mathematical Methods in Physics Keywords 35B65 35K55 35Q92 92C17 Chemotaxis Haptotaxis Quasilinear Boundedness Logistic source Industry Sectors Aerospace Engineering Oil, Gas & Geosciences Authors Ji Liu (1) Jiashan Zheng (2) Yifu Wang (1) (3) Author Affiliations 1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China 2. School of Mathematics and Information, Ludong University, Yantai, 264039, People’s Republic of China 3. Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China Continue reading... To view the rest of this content please follow the download PDF link above.