We investigate a recently proposed cross-diffusion system modeling the growth of gliobastoma taking into account size exclusion in both the migration and the proliferation process. In addition to degenerate nonlinear cross-diffusion, the model includes reaction terms as in the Fisher--Kolmogorov equation and linear ones modeling transition between states of proliferation and migration. We discuss the mathematical structure of the system and provide a complete existence analysis in spatial dimension one. The proof is based on exploiting partial entropy dissipation techniques and fully implicit time discretizations. In order to prove the existence of the latter, appropriate variational and fixed-point techniques are used, together with suitable a priori estimates. Moreover, we review the existence of traveling waves and their relation to potential growth or decay of glioblastoma. Finally, we provide extensive numerical studies in one and two spatial dimensions, including the effect of anisotropic diffusions as found in neural tissues.


  1. partial differential equation
  2. glioblastoma
  3. mathematical biology
  4. analysis
  5. numerical simulations

MSC codes

  1. 35K45
  2. 35K55
  3. 35K65
  4. 92Bxx

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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 160 - 182
ISSN (online): 1095-712X


Submitted: 15 June 2018
Accepted: 24 September 2019
Published online: 14 January 2020


  1. partial differential equation
  2. glioblastoma
  3. mathematical biology
  4. analysis
  5. numerical simulations

MSC codes

  1. 35K45
  2. 35K55
  3. 35K65
  4. 92Bxx



Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : 1073/1-2
H2020 European Research Council https://doi.org/10.13039/100010663 : 615216
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/K032208/1

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