2020/10/19 by Giao Ky Duong, Duong, G. Ky, Nikos I. Kavallaris +3 · 1 citation
Computer Science · Medicine · #14E20 #20C20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation #Primary 54C40 #Secondary 46E25
paper · pdf · doi:10.48550/arxiv.2010.09867
openalex publication_date 2020/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the current paper, we provide a thorough investigation of the blowing up\nbehaviour induced via diffusion of the solution of the following non local\nproblem \
left
∂t u · amp;= · amp;
Delta u -ν +
displaystyle
fracup
left(
mathop
,
rlap-
!
!
int
nolimits_
Omegaνr dr
right)^
gamma
quad
textin
quad
Omega
times (0,T),
\n
frac
partial u
partial
nu · amp; = · amp; 0
text on
Gamma =
partial
Omega\n
times (0,T),
u(0) · amp; = · amp; u0, \
right. where\n\Ω is a bounded domain in \ℝN with smooth boundary \∂\n\Ω; such problem is derived as the shadow limit of a singular\nGierer-Meinhardt system, cf. citeKSN17, NKMI2018. Under the Turing type\ncondition
fracrp-1 lt;
fracN2,
gamma r
ne p-1, we construct a\nsolution which blows up in finite time and only at an interior point x0 of\n\Ω, i.e. u(x0, t)
sim (
theta^*)^-
frac1p-1
left[
kappa\n(T-t)^-
frac1p-1
right], where
theta^* :=
limt
to T\n
left(
mathop
,
rlap-
!
!
int
nolimits_
Omega ur dr
right)-
gamma\n
text and
kappa = (p-1)^-
frac1p-1. More precisely, we also give a\ndescription on the final asymptotic profile at the blowup point u(x,T)
sim\n(
theta^* )^-
frac1p-1
left[
frac(p-1)28p
frac|x-x0|2\n|
ln|x-x0||
right]^ -
frac1p-1
text as x
to 0, and thus we\nunveil the form of the Turing patterns occurring in that case due to\ndriven-diffusion instability. The applied technique for the construction of the\npreceding blowing up solution mainly relies on the approach developed in\n citeMZnon97 and citeDZM3AS19.\n