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Mathematical Biosciences and Engineering

2017, Volume 14, Issue 2: 407-420. doi: 10.3934/mbe.2017025
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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
  • Received: 27 January 2016 Revised: 28 June 2016 Published: 01 April 2017

  • MSC : Primary: 35A01, 35A02, 35A09; Second: 92B05

  • This paper studies the global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with appropriate initial and mixed boundary conditions. Under some practicable regularity criteria on diffusion item and nonlinearity, we establish the local existence and uniqueness of classical solutions based on a contraction mapping. This local solution can be continued for all positive time by employing the methods of energy estimates, $ L^{p} $-theory, and Schauder estimate of linear parabolic equations. A straightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitro glioblastoma 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[J]. Mathematical Biosciences and Engineering, 2017, 14(2): 407-420. doi: 10.3934/mbe.2017025

    Related Papers:

  • This paper studies the global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with appropriate initial and mixed boundary conditions. Under some practicable regularity criteria on diffusion item and nonlinearity, we establish the local existence and uniqueness of classical solutions based on a contraction mapping. This local solution can be continued for all positive time by employing the methods of energy estimates, $ L^{p} $-theory, and Schauder estimate of linear parabolic equations. A straightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitro glioblastoma growth is also presented.


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