2019/04/08 by Monica Marras, Marras, Monica, Teruto Nishino +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1904.03856
openalex publication_date 2019/04/08 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
This paper deals with unbounded solutions to the following zero--flux\nchemotaxis system \
labelProblemAbstract
tag
Diamond\n
begincases\n % about u\n ut=
nabla
cdot [(u+
alpha)m1-1\n
nabla u-
chi u(u+
alpha)m2-2\n
nabla v]\n amp;\n (x,t)
in
Omega
times (0,Tmax),\n
\n % about v\n 0=
Delta v-M+u\n amp;\n (x,t)
in
Omega
times (0,Tmax),\n
endcases where \α>0, \Ω is a smooth and bounded\ndomain of \ℝn, with n\≥ 1, t\∈ (0, Tmax), where Tmax\nthe blow-up time, and m1,m2 real numbers. Given a sufficiently smooth\ninitial data u0:=u(x,0)\≥ 0 and set\nM:=\(1)/(|\Ω|)\∫\Ωu0(x) ,dx, from the literature it is known\nthat under a proper interplay between the above parameters m1,m2 and the\nextra condition \∫_\Ω v(x,t)dx=0, system eqrefProblemAbstract\npossesses for any \χ>0 a unique classical solution which becomes unbounded\nat t nearrow Tmax. In this investigation we first show that for\np0>\(n)/(2)(m2-m1) any blowing up classical solution in\nL^\∞(\Ω)--norm blows up also in Lp0(\Ω)--norm. Then we\nestimate the blow--up time Tmax providing a lower bound T.\n