Global well-posedness and invariant regions for a one-dimensional RD-ARD glioblastoma model
- 1Yunnan Normal Univ., Kunming, China
Abstract
We study a one-dimensional reaction-diffusion and advection-reaction-diffusion system arising from a heterogeneous glioblastoma model with shared crowding effects. The two components represent latent subpopulation densities that both diffuse and proliferate; the second is additionally subject to a constant advective bias, with crowding coupled through the total density. On a bounded interval, under either componentwise Neumann conditions or the natural zero-flux condition for the advective component, we establish a global well-posedness theory, positivity preservation, invariant regions, and uniform absorbing bounds in the supremum norm. Under the zero-flux condition, the invariant and absorbing upper barriers for the second component carry the exponential spatial weight e(a/ Dv)x, in contrast to the constant barriers available under componentwise Neumann conditions. We further formulate the associated semiflow and prove the existence of a global attractor.
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BibTeX
@article{yang2026global,
title = {{Global well-posedness and invariant regions for a one-dimensional RD-ARD glioblastoma model}},
author = {Zhipeng Yang and Zhenyu Zhan},
journal = {Electronic Journal of Differential Equations},
year = {2026},
month = sep,
doi = {10.58997/ejde.2026.68},
url = {http://ejde.math.txstate.edu/Volumes/2026/68/abstr.html},
note = {CC BY},
}
RIS
TY - JOUR
TI - Global well-posedness and invariant regions for a one-dimensional RD-ARD glioblastoma model
AU - Zhipeng Yang
AU - Zhenyu Zhan
PY - 2026
DA - 2026/09/01
JO - Electronic Journal of Differential Equations
DO - 10.58997/ejde.2026.68
UR - http://ejde.math.txstate.edu/Volumes/2026/68/abstr.html
AB - We study a one-dimensional reaction-diffusion and advection-reaction-diffusion system arising from a heterogeneous glioblastoma model with shared crowding effects. The two components represent latent subpopulation densities that both diffuse and proliferate; the second is additionally subject to a constant advective bias, with crowding coupled through the total density. On a bounded interval, under either componentwise Neumann conditions or the natural zero-flux condition for the advective component, we establish a global well-posedness theory, positivity preservation, invariant regions, and uniform absorbing bounds in the supremum norm. Under the zero-flux condition, the invariant and absorbing upper barriers for the second component carry the exponential spatial weight e^(a/ D_v)x, in contrast to the constant barriers available under componentwise Neumann conditions. We further formulate the associated semiflow and prove the existence of a global attractor.
SN - 1072-6691
ER -