2019/03/19 by Giuseppe Viglialoro, Viglialoro, Giuseppe
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth
paper · pdf · doi:10.48550/arxiv.1903.08196
openalex publication_date 2019/03/19 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
In this paper we study classical solutions to the zero--flux\nattraction--repulsion chemotaxis--system\n\
labelProblemAbstract
tag
Diamond
begincases u\nt=
Delta u -
chi
nabla
cdot (u
nabla v)+
xi
nabla
cdot (u
nabla w) amp;\n
textrmin
Omega
times (0,t^*),
0=
Delta v+
alpha u-
beta v amp;
textrmin \n
Omega
times (0,t^*),
0=
Delta w+
gamma u-
delta w amp;
textrmin \n
Omega
times (0,t^*),
endcases where \Ω is a smooth\nand bounded domain of \ℝ2, t^* is the blow--up time and\n\α,\β,\γ,\δ,\χ,\ξ are positive real numbers. From the\nliterature it is known that under a proper interplay between the above\nparameters and suitable smallness assumptions on the initial data u( bf\nx,0)=u0\∈ C0(\\Ω), system eqrefProblemAbstract has a unique\nclassical solution which becomes unbounded as t nearrow t^*. The main result\nof this investigation is to provide an explicit lower bound for t^* estimated\nin terms of \∫_\Ω u02 d bf x and attained by means of\nwell--established techniques based on ordinary differential inequalities.\n