Advanced Search
Article Contents
Article Contents

Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model

1Meng Fan is partially supported by NSFC-11271065, RFPD-20130043110001, and RFCPCMSP-2014

Abstract Full Text(HTML) Related Papers Cited by
  • This paper studies the global existence and uniqueness of classicalsolutions for a generalized quasilinear parabolic equation withappropriate initial and mixed boundary conditions. Under somepracticable regularity criteria on diffusion item and nonlinearity, weestablish the local existence and uniqueness of classical solutionsbased on a contraction mapping. This local solution can be continuedfor all positive time by employing the methods of energy estimates, $ L^{p} $-theory, and Schauder estimate of linear parabolic equations. Astraightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitroglioblastoma growth is also presented.

    Mathematics Subject Classification: Primary: 35A01, 35A02, 35A09; Second: 92B05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] M. Agueh, Gagliardo-Nirenberg inequalities involving the gradient $ L^{2} $-norm, C. R. Acad. Sci. Paris, Ser., 346 (2008), 757-762.  doi: 10.1016/j.crma.2008.05.015.
    [2] H. Amann, Dynamic theory of quasilinear parabolic equations-Ⅰ. Abstract evolution equations, Nonlinear Anal., 12 (1988), 895-919.  doi: 10.1016/0362-546X(88)90073-9.
    [3] H. Amann, Dynamic theory of quasilinear parabolic equations-Ⅲ. Global existence, Math. Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.
    [4] H. Amann, Dynamic theory of quasilinear parabolic equations-Ⅰ. Reaction-diffusion, Diff. Int. Eqs, 3 (1990), 13-75. 
    [5] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.
    [6] M. Bause and K. Schwegler, Analysis of stabilized higher-order finite element approximation of nonstationary and nonlinear convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Engrg., 209/212 (2012), 184-196.  doi: 10.1016/j.cma.2011.10.004.
    [7] A. Q. CaiK. A. Landman and B. D. Hughes, Multi-scale modeling of a wound-healing cell migration assay, J. Theor. Biol., 245 (2007), 576-594.  doi: 10.1016/j.jtbi.2006.10.024.
    [8] B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Vol. 362,2004. doi: 10.1007/978-3-0348-7964-4.
    [9] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.
    [10] V. John and E. Schmeyer, On finite element methods for 3D time-dependent convectiondiffusion-reaction equations with small diffusion, BAIL 2008-Boundary and Interior Layers, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 69 (2009), 173-181. doi: 10.1007/978-3-642-00605-0_13.
    [11] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Amer. Math. Soc., Vol. 23,1968.
    [12] J. M. LeeT. Hillena and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dynamics, 3 (2009), 551-573.  doi: 10.1080/17513750802716112.
    [13] G. P. Mailly and J. F. Rault, Nonlinear convection in reaction-diffusion equations under dynamical boundary conditions, Electronic J. Diff. Eqns, 2013 (2013), 1-14. 
    [14] J. D. Murray, Mathematical Biology Ⅰ: An Introduction Springer, Vol. 17,2002, $ 3^{rd} $ Edition.
    [15] H. G. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.  doi: 10.1137/S0036139995288976.
    [16] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. 
    [17] C. V. Pao and W. H. Ruan, Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition, J. Diff. Eqns, 248 (2011), 1175-1211.  doi: 10.1016/j.jde.2009.12.011.
    [18] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford University Press, 1997.
    [19] T. L. StepienE. M. Rutter and Y. Kuang, A data-motivated density-dependent diffusion model of in vitro glioblastoma growth, Mathematical Biosciences and Engineering, 12 (2015), 1157-1172.  doi: 10.3934/mbe.2015.12.1157.
    [20] Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonl. Aanl.: RWA, 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.
    [21] Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Diff. Eqns., 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.
    [22] Z. Yin, On the global existence of solutions to quasilinear parabolic equations with homogeneous Neumann boundary conditions, Glasgow Math. J., 47 (2005), 237-248.  doi: 10.1017/S0017089505002442.
  • 加载中

Article Metrics

HTML views(192) PDF downloads(199) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint