# American Institute of Mathematical Sciences

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. Google Scholar [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. Google Scholar [3] H. Amann, Dynamic theory of quasilinear parabolic equations-Ⅲ. Global existence, Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256. Google Scholar [4] H. Amann, Dynamic theory of quasilinear parabolic equations-Ⅰ. Reaction-diffusion, Diff. Int. Eqs, 3 (1990), 13-75. Google Scholar [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. Google Scholar [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. Google Scholar [7] A. Q. Cai, K. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] J. M. Lee, T. Hillena and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dynamics, 3 (2009), 551-573. doi: 10.1080/17513750802716112. Google Scholar [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. Google Scholar [14] J. D. Murray, Mathematical Biology Ⅰ: An Introduction Springer, Vol. 17,2002, $3^{rd}$ Edition. Google Scholar [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. Google Scholar [16] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. Google Scholar [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. Google Scholar [18] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford University Press, 1997.Google Scholar [19] T. L. Stepien, E. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [3] H. Amann, Dynamic theory of quasilinear parabolic equations-Ⅲ. Global existence, Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256. Google Scholar [4] H. Amann, Dynamic theory of quasilinear parabolic equations-Ⅰ. Reaction-diffusion, Diff. Int. Eqs, 3 (1990), 13-75. Google Scholar [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. Google Scholar [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. Google Scholar [7] A. Q. Cai, K. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] J. M. Lee, T. Hillena and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dynamics, 3 (2009), 551-573. doi: 10.1080/17513750802716112. Google Scholar [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. Google Scholar [14] J. D. Murray, Mathematical Biology Ⅰ: An Introduction Springer, Vol. 17,2002, $3^{rd}$ Edition. Google Scholar [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. Google Scholar [16] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375. Google Scholar [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. Google Scholar [18] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford University Press, 1997.Google Scholar [19] T. L. Stepien, E. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
 [1] Tracy L. Stepien, Erica M. Rutter, Yang Kuang. A data-motivated density-dependent diffusion model of in vitro glioblastoma growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1157-1172. doi: 10.3934/mbe.2015.12.1157 [2] Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207 [3] Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209 [4] Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41 [5] Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395 [6] Jacques A. L. Silva, Flávia T. Giordani. Density-dependent dispersal in multiple species metapopulations. Mathematical Biosciences & Engineering, 2008, 5 (4) : 843-857. doi: 10.3934/mbe.2008.5.843 [7] Jishan Fan, Tohru Ozawa. Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model. Conference Publications, 2011, 2011 (Special) : 400-409. doi: 10.3934/proc.2011.2011.400 [8] Tianyuan Xu, Shanming Ji, Chunhua Jin, Ming Mei, Jingxue Yin. Early and late stage profiles for a chemotaxis model with density-dependent jump probability. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1345-1385. doi: 10.3934/mbe.2018062 [9] Chuangxia Huang, Hua Zhang, Lihong Huang. Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3337-3349. doi: 10.3934/cpaa.2019150 [10] Francisco Guillén-González, Mamadou Sy. Iterative method for mass diffusion model with density dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 823-841. doi: 10.3934/dcdsb.2008.10.823 [11] Pierre Degond, Silke Henkes, Hui Yu. Self-organized hydrodynamics with density-dependent velocity. Kinetic & Related Models, 2017, 10 (1) : 193-213. doi: 10.3934/krm.2017008 [12] J. X. Velasco-Hernández, M. Núñez-López, G. Ramírez-Santiago, M. Hernández-Rosales. On carrying-capacity construction, metapopulations and density-dependent mortality. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1099-1110. doi: 10.3934/dcdsb.2017054 [13] Baojun Song, Wen Du, Jie Lou. Different types of backward bifurcations due to density-dependent treatments. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1651-1668. doi: 10.3934/mbe.2013.10.1651 [14] Gui-Qiang Chen, Kenneth Hvistendahl Karlsen. Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Communications on Pure & Applied Analysis, 2005, 4 (2) : 241-266. doi: 10.3934/cpaa.2005.4.241 [15] Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 [16] Quansen Jiu, Zhouping Xin. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinetic & Related Models, 2008, 1 (2) : 313-330. doi: 10.3934/krm.2008.1.313 [17] Jianwei Yang, Peng Cheng, Yudong Wang. Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. Electronic Research Announcements, 2015, 22: 20-31. doi: 10.3934/era.2015.22.20 [18] Azmy S. Ackleh, Linda J. S. Allen. Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 175-188. doi: 10.3934/dcdsb.2005.5.175 [19] Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure & Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373 [20] Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041

2018 Impact Factor: 1.313