April  2017, 14(2): 407-420. doi: 10.3934/mbe.2017025

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

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

2. 

School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, China

3. 

Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

4. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA

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

Received  January 27, 2016 Revised  June 28, 2016 Published  August 2016

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.

Citation: Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang. Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 407-420. doi: 10.3934/mbe.2017025
References:
[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.

show all references

References:
[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.

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