Boundedness in a quasilinear chemotaxis–haptotaxis system with logistic source

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• 作者：Ji Liu ; Jiashan Zheng ; Yifu Wang
• 关键词：Chemotaxis ; Haptotaxis ; Quasilinear ; Boundedness ; Logistic source
• 刊名：Zeitschrift f¨¹r angewandte Mathematik und Physik
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：67
• 期：2
• 全文大小：801 KB
• 参考文献：1.Alikakos N.D.: Bounds of solutions of reaction-diffusion equations. Commun. Partial Differ. Eqn. 4, 827–868 (1979)MathSciNet CrossRef MATH
  2.Biler P.: Local and global solvability of some parabolic systems modelling chemotaxis. Adv. Math. Sci. Appl. 8, 715–743 (1998)MathSciNet MATH
  3.Calvez V., Carrillo J.A.: Volume effects in the Keller-Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. 86(9), 155–175 (2006)MathSciNet CrossRef MATH
  4.Cao, X.: Boundedness in a three-dimensional chemotaxis-haptotaxis model. arXiv:​1501.​05383 (2015)
  5.Chaplain M.A.J., Anderson A.R.A.: Mathematical modelling of tissue invasion. In: Preziosi, L. (eds) Cancer Modelling and Simulation, pp. 269–297. Chapman & Hall/CRC, Boca Raton (2003)
  6.Chaplain M.A.J., Lolas G.: Mathematical modelling of cancer invasion of tissue: the role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 15, 1685–1734 (2005)MathSciNet CrossRef MATH
  7.Cieślak T., Laurenco̧t P.: Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski–Poisson system. C. R. Acad. Sci. Paris 347, 237–242 (2009)MathSciNet CrossRef MATH
  8.Cieślak T., Winkler M.: Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity 21, 1057–1076 (2008)MathSciNet CrossRef MATH
  9.Djie K., Winkler M.: Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. Nonlinear Anal. TMA 72, 1044–1064 (2010)MathSciNet CrossRef MATH
  10.Herrero M.A., Velázquez J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. Pisa Cl. Sci. 24(4), 633–683 (1997)MathSciNet MATH
  11.Hillen T., Painter K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)MathSciNet CrossRef MATH
  12.Hillen T., Painter K.J.: Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math. 26, 281–301 (2001)MathSciNet CrossRef MATH
  13.Horstmann D., Wang G.: Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12, 159–177 (2001)MathSciNet CrossRef MATH
  14.Horstmann D., Winkler M.: Boundedness vs. blow-up in a chemotaxis system. J. Diff. Eqn. 215(1), 52–107 (2005)MathSciNet CrossRef MATH
  15.Ishida S., Seki K., Yokota T.: Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains. J. Diff. Eqn. 256, 2993–3010 (2014)MathSciNet CrossRef MATH
  16.Jäger W., Luckhaus S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992)MathSciNet CrossRef MATH
  17.Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)CrossRef MATH
  18.Kowalczyk R.: Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305, 566–585 (2005)MathSciNet CrossRef MATH
  19.Kowalczyk R., Szymańska Z.: On the global existence of solutions to an aggregation model. J. Math. Anal. Appl. 343, 379–398 (2008)MathSciNet CrossRef MATH
  20.Ladyzenskaja O.A., Solonnikov V.A., Ural’ceva N.N.: Linear and Quasi-Linear Equations of Parabolic Type. AMS, Providence, RI (1968)
  21.Liţcanu G., Morales-Rodrigo C.: Asymptotic behavior of global solutions to a model of cell invasion. Math. Models Methods Appl. Sci. 20, 1721–1758 (2010)MathSciNet CrossRef MATH
  22.Marciniak-Czochra A., Ptashnyk M.: Boundedness of solutions of a haptotaxis model. Math. Models Methods Appl. Sci. 20, 449–476 (2010)MathSciNet CrossRef MATH
  23.Mizoguchi N., Souplet P.: Nondegeneracy of blow-up points for the parabolic Keller–Segel system. Ann. Inst. H. Poincaré Anal. Non Linéaire Anal. 31, 851–875 (2014)MathSciNet CrossRef MATH
  24.Nagai T.: Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. J. Inequal. Appl. 6, 37–55 (2001)MathSciNet MATH
  25.Osaki K., Tsujikawa T., Yagi A., Mimura M.: Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. TMA 51, 119–144 (2002)MathSciNet CrossRef MATH
  26.Painter K.J., Hillen T.: Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Q. 10, 501–543 (2002)MathSciNet MATH
  27.Perthame B.: Transport Equations in Biology. Birkhäuser Verlag, Basel (2007)MATH
  28.Perumpanani A.J., Byrne H.M.: Extracellular matrix concentration exerts selection pressure on invasive cells. Eur. J. Cancer 35, 1274–1280 (1999)CrossRef
  29.Rascle M., Ziti C.: Finite time blow-up in some models of chemotaxis. J. Math. Biol. 33, 388–414 (1995)MathSciNet CrossRef MATH
  30.Sugiyama Y.: Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic systems of chemotaxis. Differ. Integral Equ. 20, 133–180 (2007)MathSciNet MATH
  31.Szymańska Z., Morales-Rodrigo C., Lachowicz M., Chaplain M.: Mathematical modelling of cancer invasion of tissue: the role and effect of nonlocal interactions. Math. Models Methods Appl. Sci. 19, 257–281 (2009)MathSciNet CrossRef MATH
  32.Tao Y.: Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source. J. Math. Anal. Appl. 354, 60–69 (2009)MathSciNet CrossRef MATH
  33.Tao, Y.: Boundedness in a two-dimensional chemotaxis-haptotaxis system. arXiv:​1407.​7382 (2014)
  34.Tao Y., Wang M.: Global solution for a chemotactic-haptotactic model of cancer invasion. Nonlinearity 21, 2221–2238 (2008)MathSciNet CrossRef MATH
  35.Tao Y., Winkler M.: A chemotaxis-haptotaxis model: the roles of nonlinear diffusion and logistic source. SIAM J. Math. Anal. 43, 685–704 (2011)MathSciNet CrossRef MATH
  36.Tao Y., Winkler M.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Diff. Eqn. 252, 692–715 (2012)MathSciNet CrossRef MATH
  37.Tao Y., Winkler M.: Dominance of chemotaxis in a chemotaxis–haptotaxis model. Nonlinearity 27, 1225–1239 (2014)MathSciNet CrossRef MATH
  38.Tello J.I.: Mathematical analysis and stability of a chemotaxis model with logistic term. Math. Methods Appl. Sci. 27, 1865–1880 (2004)MathSciNet CrossRef MATH
  39.Tello J.I., Winkler M.: A chemotaxis system with logistic source. Commun. Partial Diff. Eqn. 32, 849–877 (2007)MathSciNet CrossRef MATH
  40.Wang L., Li Y., Mu C.: Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete Continuous Dyn. Syst. Ser. A 34, 789–802 (2014)MathSciNet CrossRef MATH
  41.Winkler M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J. Diff. Eqn. 248, 2889–2905 (2010)MathSciNet CrossRef MATH
  42.Winkler M.: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Commun. Partial Diff. Eqn. 35, 1516–1537 (2010)MathSciNet CrossRef MATH
  43.Winkler M.: Chemotaxis with logistic source: very weak global solutions and their boundedness properties. J. Math. Anal. Appl. 348, 708–729 (2008)MathSciNet CrossRef MATH
  44.Winkler M., Djie K.C.: Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. Nonlinear Anal. TMA 72, 1044–1064 (2010)MathSciNet CrossRef MATH
  45.Wrzosek D.: Global attractor for a chemotaxis model with prevention of overcrowding. Nonlinear Anal. TMA 59, 1293–1310 (2004)MathSciNet CrossRef MATH
  
• 作者单位：Ji Liu (1)
  Jiashan Zheng (2)
  Yifu Wang (1) (3)
  
  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
  
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer Basel
• ISSN：1420-9039

文摘

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. Pures Appl. 86(9), 155–175 (2006)MathSciNetCrossRefMATH4.Cao, X.: Boundedness in a three-dimensional chemotaxis-haptotaxis model. arXiv:​1501.​05383 (2015)5.Chaplain M.A.J., Anderson A.R.A.: Mathematical modelling of tissue invasion. In: Preziosi, L. (eds) Cancer Modelling and Simulation, pp. 269–297. Chapman & Hall/CRC, Boca Raton (2003)6.Chaplain M.A.J., Lolas G.: Mathematical modelling of cancer invasion of tissue: the role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 15, 1685–1734 (2005)MathSciNetCrossRefMATH7.Cieślak T., Laurenco̧t P.: Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski–Poisson system. C. R. Acad. Sci. Paris 347, 237–242 (2009)MathSciNetCrossRefMATH8.Cieślak T., Winkler M.: Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity 21, 1057–1076 (2008)MathSciNetCrossRefMATH9.Djie K., Winkler M.: Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. Nonlinear Anal. TMA 72, 1044–1064 (2010)MathSciNetCrossRefMATH10.Herrero M.A., Velázquez J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. Pisa Cl. Sci. 24(4), 633–683 (1997)MathSciNetMATH11.Hillen T., Painter K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)MathSciNetCrossRefMATH12.Hillen T., Painter K.J.: Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math. 26, 281–301 (2001)MathSciNetCrossRefMATH13.Horstmann D., Wang G.: Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12, 159–177 (2001)MathSciNetCrossRefMATH14.Horstmann D., Winkler M.: Boundedness vs. blow-up in a chemotaxis system. J. Diff. Eqn. 215(1), 52–107 (2005)MathSciNetCrossRefMATH15.Ishida S., Seki K., Yokota T.: Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains. J. Diff. Eqn. 256, 2993–3010 (2014)MathSciNetCrossRefMATH16.Jäger W., Luckhaus S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992)MathSciNetCrossRefMATH17.Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)CrossRefMATH18.Kowalczyk R.: Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305, 566–585 (2005)MathSciNetCrossRefMATH19.Kowalczyk R., Szymańska Z.: On the global existence of solutions to an aggregation model. J. Math. Anal. Appl. 343, 379–398 (2008)MathSciNetCrossRefMATH20.Ladyzenskaja O.A., Solonnikov V.A., Ural’ceva N.N.: Linear and Quasi-Linear Equations of Parabolic Type. AMS, Providence, RI (1968)21.Liţcanu G., Morales-Rodrigo C.: Asymptotic behavior of global solutions to a model of cell invasion. Math. Models Methods Appl. Sci. 20, 1721–1758 (2010)MathSciNetCrossRefMATH22.Marciniak-Czochra A., Ptashnyk M.: Boundedness of solutions of a haptotaxis model. Math. Models Methods Appl. Sci. 20, 449–476 (2010)MathSciNetCrossRefMATH23.Mizoguchi N., Souplet P.: Nondegeneracy of blow-up points for the parabolic Keller–Segel system. Ann. Inst. H. Poincaré Anal. Non Linéaire Anal. 31, 851–875 (2014)MathSciNetCrossRefMATH24.Nagai T.: Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. J. Inequal. Appl. 6, 37–55 (2001)MathSciNetMATH25.Osaki K., Tsujikawa T., Yagi A., Mimura M.: Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. TMA 51, 119–144 (2002)MathSciNetCrossRefMATH26.Painter K.J., Hillen T.: Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Q. 10, 501–543 (2002)MathSciNetMATH27.Perthame B.: Transport Equations in Biology. Birkhäuser Verlag, Basel (2007)MATH28.Perumpanani A.J., Byrne H.M.: Extracellular matrix concentration exerts selection pressure on invasive cells. Eur. J. Cancer 35, 1274–1280 (1999)CrossRef29.Rascle M., Ziti C.: Finite time blow-up in some models of chemotaxis. J. Math. Biol. 33, 388–414 (1995)MathSciNetCrossRefMATH30.Sugiyama Y.: Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic systems of chemotaxis. Differ. Integral Equ. 20, 133–180 (2007)MathSciNetMATH31.Szymańska Z., Morales-Rodrigo C., Lachowicz M., Chaplain M.: Mathematical modelling of cancer invasion of tissue: the role and effect of nonlocal interactions. Math. Models Methods Appl. Sci. 19, 257–281 (2009)MathSciNetCrossRefMATH32.Tao Y.: Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source. J. Math. Anal. Appl. 354, 60–69 (2009)MathSciNetCrossRefMATH33.Tao, Y.: Boundedness in a two-dimensional chemotaxis-haptotaxis system. arXiv:​1407.​7382 (2014)34.Tao Y., Wang M.: Global solution for a chemotactic-haptotactic model of cancer invasion. Nonlinearity 21, 2221–2238 (2008)MathSciNetCrossRefMATH35.Tao Y., Winkler M.: A chemotaxis-haptotaxis model: the roles of nonlinear diffusion and logistic source. SIAM J. Math. Anal. 43, 685–704 (2011)MathSciNetCrossRefMATH36.Tao Y., Winkler M.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Diff. Eqn. 252, 692–715 (2012)MathSciNetCrossRefMATH37.Tao Y., Winkler M.: Dominance of chemotaxis in a chemotaxis–haptotaxis model. Nonlinearity 27, 1225–1239 (2014)MathSciNetCrossRefMATH38.Tello J.I.: Mathematical analysis and stability of a chemotaxis model with logistic term. Math. Methods Appl. Sci. 27, 1865–1880 (2004)MathSciNetCrossRefMATH39.Tello J.I., Winkler M.: A chemotaxis system with logistic source. Commun. Partial Diff. Eqn. 32, 849–877 (2007)MathSciNetCrossRefMATH40.Wang L., Li Y., Mu C.: Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete Continuous Dyn. Syst. Ser. A 34, 789–802 (2014)MathSciNetCrossRefMATH41.Winkler M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J. Diff. Eqn. 248, 2889–2905 (2010)MathSciNetCrossRefMATH42.Winkler M.: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Commun. Partial Diff. Eqn. 35, 1516–1537 (2010)MathSciNetCrossRefMATH43.Winkler M.: Chemotaxis with logistic source: very weak global solutions and their boundedness properties. J. Math. Anal. Appl. 348, 708–729 (2008)MathSciNetCrossRefMATH44.Winkler M., Djie K.C.: Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. Nonlinear Anal. 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. 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