Preprint series: 04-24 , Reports on Analysis and Numerical Mathematics
Abstract: 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 H'older
regularity results for the diffusion equation.
Keywords: taxis-diffusion-reaction models, vanishing motility, hyperbolic equations, method of characteristics, nonlinear parabolic equations, maximal regularity
Notes: Submitted for publication.
Upload: 2004-08-31