Well-posedness of a quasilinear hyperbolic-parabolic system arising in mathematical biology

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2007
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Gerisch, Alf
Zacher, Rico
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Nonlinear Differential Equations and Applications NoDEA. 2007, 14(5-6), pp. 593-624. ISSN 1021-9722. eISSN 1420-9004. Available under: doi: 10.1007/s00030-007-5023-2
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We study the existence of classical solutions of a taxis-diffusion-reaction model for tumour-induced blood vessel growth. The model in its basic form has been proposed by Chaplain and Stuart (IMA J. Appl. Med. Biol. (10), 1993) and consists of one equation for the endothelial cell-density and another one for the concentration of tumour angiogenesis factor (TAF). Here we consider the special and interesting case that endothelial cells are immobile in the absence of TAF, i.e. vanishing cell motility. In this case the mathematical structure of the model changes significantly (from parabolic type to a mixed hyperbolic-parabolic type) and existence of solutions is by no means clear. We present conditions on the initial and boundary data which guarantee local existence, uniqueness and positivity of classical solutions of the problem. Our approach is based on the method of characteristics and relies on known maximal L p and Hölder regularity results for the diffusion equation.

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510 Mathematik
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ISO 690GERISCH, Alf, Matthias KOTSCHOTE, Rico ZACHER, 2007. Well-posedness of a quasilinear hyperbolic-parabolic system arising in mathematical biology. In: Nonlinear Differential Equations and Applications NoDEA. 2007, 14(5-6), pp. 593-624. ISSN 1021-9722. eISSN 1420-9004. Available under: doi: 10.1007/s00030-007-5023-2
BibTex
@article{Gerisch2007Wellp-25496,
  year={2007},
  doi={10.1007/s00030-007-5023-2},
  title={Well-posedness of a quasilinear hyperbolic-parabolic system arising in mathematical biology},
  number={5-6},
  volume={14},
  issn={1021-9722},
  journal={Nonlinear Differential Equations and Applications NoDEA},
  pages={593--624},
  author={Gerisch, Alf and Kotschote, Matthias and Zacher, Rico}
}
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    <dcterms:abstract xml:lang="eng">We study the existence of classical solutions of a taxis-diffusion-reaction model for tumour-induced blood vessel growth. The model in its basic form has been proposed by Chaplain and Stuart (IMA J. Appl. Med. Biol. (10), 1993) and consists of one equation for the endothelial cell-density and another one for the concentration of tumour angiogenesis factor (TAF). Here we consider the special and interesting case that endothelial cells are immobile in the absence of TAF, i.e. vanishing cell motility. In this case the mathematical structure of the model changes significantly (from parabolic type to a mixed hyperbolic-parabolic type) and existence of solutions is by no means clear. We present conditions on the initial and boundary data which guarantee local existence, uniqueness and positivity of classical solutions of the problem. Our approach is based on the method of characteristics and relies on known maximal L p and Hölder regularity results for the diffusion equation.</dcterms:abstract>
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