Global weak solutions and asymptotics of a singular PDE-ODE Chemotaxis system with discontinuous data
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https://doi.org/10.1007/s11425-019-1754-0
Global weak solutions and asymptotics of a singular PDE-ODE chemotaxis system with discontinuous data Hongyun Peng1 , Zhi-An Wang2,∗ & Changjiang Zhu3 1School
of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China; of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong 999077, China; 3School of Mathematics, South China University of Technology, Guangzhou 510641, China
2Department
Email: [email protected], [email protected], [email protected] Received March 19, 2020; accepted July 31, 2020
Abstract
This paper is concerned with the well-posedness and large-time behavior of a two-dimensional PDE-
ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We first transform the system via a Cole-Hopf type transformation into a parabolic-hyperbolic system and then show that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. Finally we reverse the Cole-Hopf transformation and obtain the relevant results for the pre-transformed PDE-ODE hybrid system. In contrast to the existing related results, where continuous initial data is imposed, we are able to prove the asymptotic stability for discontinuous initial data with large oscillations. The key ingredient in our proof is the use of the so-called “effective viscous flux”, which enables us to obtain the desired energy estimates and regularity. The technique of the “effective viscous flux” turns out to be a very useful tool to study chemotaxis systems with initial data having low regularity and was rarely (if not) used in the literature for chemotaxis models. Keywords MSC(2010)
chemotaxis, asymptotic stability, discontinuous initial data, effective viscous flux 35A01, 35B40, 35Q92, 92C17
Citation: Peng H Y, Wang Z-A, Zhu C J. Global weak solutions and asymptotics of a singular PDE-ODE chemotaxis system with discontinuous data. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-0191754-0
1
Introduction
In this paper, we study the following PDE-ODE hybrid chemotaxis model: { ut = ∆u − ∇ · (ξu∇ ln c),
(1.1)
ct = −µuc
which was proposed in [28] to describe the interaction between the signaling molecules vascular endothelial growth factor (VEGF) and vascular endothelial cells during the initiation of tumor angiogenesis (see * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
math.scichina.com
link.springer.com
2
Peng H Y et al.
Sci China Math
also [5, 6]). Here, u(x, t) denotes the density of vascular endothelial cells and c(x, t) means the concentration of the VEGF. This parameter ξ > 0 indicates the chemotactic coefficient measuring the strength of chemotaxis and µ is the degradation rate of the chemical VEGF. Here, chemical diffusion is ignored because it is far less important than its interaction with endothelial cells (see [28]). On the other hand, this system can also be regarded as a special form of the famou
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