2016/05/06 by Nicola Bellomo, Michael Winkler, Bellomo, Nicola +1 · 2 citations
Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #MRI in cancer diagnosis #Mathematical Biology Tumor Growth
paper · pdf · doi:10.48550/arxiv.1605.01924
openalex publication_date 2016/05/06 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
This paper aims at providing a first step toward a qualitative theory for a\nnew class of chemotaxis models derived from the celebrated Keller-Segel system,\nwith the main novelty being that diffusion is nonlinear with flux delimiter\nfeatures. More precisely, as a prototypical representative of this class we\nstudy radially symmetric solutions of the parabolic-elliptic system (see the\ntext).\n Under the initial condition u|t=0=u0>0 and no-flux boundary conditions\nin balls \Ω\⊂\ℝn, where \χ>0 and\n\μ:=\(1)/(|\Ω|) \∫_\Ω u0. abs\n The main results assert the existence of a unique classical solution,\nextensible in time up to a maximal Tmax \∈ (0,\∞] which has the\nproperty that
mboxif
quad Tmaxlt;
infty
quad
mboxthen\n
quad
limsup_t
nearrow Tmax
|u(
cdot,t)
|L^
infty(
Omega)=
infty.\n
qquad
qquad (
star)\n The proof therefore is mainly based on comparison methods, which firstly\nrelate pointwise lower and upper bounds for the spatial gradient ur to\nL^\∞ bounds for u and to em upper bounds for z:=\(ut)/(u);\nsecondly, another comparison argument involving nonlocal nonlinearities\nprovides an appropriate control of z+ in terms of bounds for u and\n|ur|, with suitably mild dependence on the latter.\n As a consequence of (\⋆) by means of suitable a priori estimates it is\nmoreover shown that the above solutions are global and bounded when either\nn
ge 2
mbox and
chilt;1,
qquad
mboxor
qquad n=1,
chigt;0
mbox\nand mlt;mc, with mc:=\(1)/(\√(\χ2-1)) if \χ>1 and\nmc:=\∞ if \χ\≤ 1. That these conditions are essentially optimal\nwill be shown in a forthcoming paper in which (\⋆) will be used to derive\ncomplementary results on the occurrence of solutions blowing up in finite time\nwith respect to the norm of u in L^\∞(\Ω).\n