2019/03/14 by Nancy Rodríguez, Michael Winkler, Rodriguez, Nancy +1 · 1 citation
Mathematics · Medicine · #35B40 #35K55 (secondary) #35Q91 (primary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1903.06331
openalex publication_date 2019/03/14 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We consider the no-flux initial-boundary value problem for the\ncross-diffusive evolution system n
left
n ut = uxx -
chi
big(
fracuv
partialx u
big)x - uv +B1(x,t),\n
qquad · amp; x
in
Omega,
t · gt;0,
\n vt = vxx +uv - v + B2(x,t),\n
qquad · amp; x
in
Omega,
t · gt;0,\n \
right. n which was introduced by Short et al. in [Short2008] with \χ=2 to describe\nthe dynamics of urban crime\n In bounded intervals \Ω\⊂\ℝ and with prescribed suitably\nregular nonnegative\n functions B1 and B2, we first prove the existence of global classical\nsolutions for any choice of \χ>0 and all reasonably\n regular nonnegative initial data. We next address the issue of determining\nthe qualitative behavior of solutions under appropriate assumptions\n on the asymptotic properties of B1 and B2. Indeed, for arbitrary\n\χ>0 we obtain boundedness of the solutions given strict positivity of the\naverage of B2\n over the domain; moreover, it is seen that imposing a mild decay assumption\non B1 implies that u must\n decay to zero in the long-term limit.\n Our final result, valid for all\n\χ\∈\(0,\(\√(6\√(3)+9))/(2)\), which contains\n the relevant value \χ=2, states that under the above decay assumption on\nB1, if furthermore B2 appropriately stabilizes to a\n nontrivial function B2,\∞, then (u,v) approaches the limit\n(0,v_\∞), where v_\∞\n denotes the solution of n
left
n -
partialxxv_
infty + v_
infty = B2,
infty,\n
qquad x
in
Omega,
\n
partialx v
infty=0,\n
qquad x
in
partial
Omega.\n \
right. n