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Combined effects of nonlinear diffusion and gradient-dependent flux limitation on a chemotaxis–haptotaxis model

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Zhan Jiao
Zhan Jiao
Zhan Jiao
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Irena Jadlovska at Slovak Academy of Sciences
Irena Jadlovska
Tongxing Li
Tongxing Li
Tongxing Li
  • This person is not on ResearchGate, or hasn't claimed this research yet.
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Abstract

The flux-limited chemotaxis–haptotaxis system is considered in a smooth and bounded domain \(\Omega \subset \mathbb R^n\) \((n\ge 2)\), where \(\tau \in \{0,1\}\), \(\chi ,\xi \) and \(\eta \) are positive parameters, as well as the functions D, f and h satisfy \(D(u)\ge d(u+1)^{m-1}\), \(f(|\nabla v|)\le |\nabla v|^{p-2}\) and \( h(u,w)=u(a-\mu u^{r-1}-\lambda w)\) with \(a\in \mathbb R\), \(d,\mu ,\lambda >0\), \(r>1\), \(r\ge 2\eta \), \(m\ge 1\) and \(1<p<\frac{n}{n-1}\). Firstly, for all reasonably regular initial data, we confirm the existence of a globally defined bounded classical solution if either \(\tau =1\) and \(n\in \{2,3\}\) or \(\tau =0\) and \(n\ge 2\). Moreover, when \(\tau =0\) and alternatively \(h(u,w)=a+au-\mu u^r-\lambda uw\) with \(a>0\) and \(r\ge 2\), it turns out that exponential decay of w is detected on large time scales, while both u and v persist in a certain manner for higher dimensions, provided that \(\eta \ge 1\), \(r\ge 2\eta \) and \(\mu >\frac{\eta \chi ^2}{4r}\).
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Z. Angew. Math. Phys. (2024) 75:4
c
2023 The Author(s), under exclusive licence to Springer Nature
Switzerland AG
0044-2275/24/010001-19
published online December 8, 2023
https://doi.org/10.1007/s00033-023-02134-2
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Combined effects of nonlinear diffusion and gradient-dependent flux limitation
on a chemotaxis–haptotaxis model
Zhan Jiao, Irena Jadlovsk´a and Tongxing Li
Abstract. The flux-limited chemotaxis–haptotaxis system
⎧
⎪
⎨
⎪
⎩
ut=∇·(D(u)∇u)−χ∇·(uf(|∇v|)∇v)−ξ∇·(u∇w)+h(u, w),x∈Ω,t>0,
τv
t=Δv−v+uη,x∈Ω,t>0,
wt=−vw, x ∈Ω,t>0
is considered in a smooth and bounded domain Ω ⊂Rn(n≥2), where τ∈{0,1},χ, ξ and ηare positive parameters, as
well as the functions D, f and hsatisfy D(u)≥d(u+1)
m−1,f(|∇v|)≤|∇v|p−2and h(u, w)=u(a−μur−1−λw)with
a∈R,d, μ, λ > 0, r>1, r≥2η,m≥1and1<p< n
n−1. Firstly, for all reasonably regular initial data, we confirm the
existence of a globally defined bounded classical solution if either τ=1andn∈{2,3}or τ=0andn≥2. Moreover, when
τ= 0 and alternatively h(u, w )=a+au −μur−λuw with a>0andr≥2, it turns out that exponential decay of wis
detected on large time scales, while both uand vpersist in a certain manner for higher dimensions, provided that η≥1,
r≥2ηand μ> ηχ2
4r.
Mathematics Subject Classification. 35A01, 35B35, 35K55, 35Q92, 92C17.
Keywords. Chemotaxis–haptotaxis, Boundedness, Asymptotic stability, Logistic source, Flux limitation.
1. Introduction
Chemotaxis refers to the phenomenon where the movement direction of biological agents adapts in re-
sponse to external chemical signals that can either be produced or consumed by these agents themselves.
This concept was originally proposed in the seminal work of Keller and Segel [11]. Since then, the classical
chemotaxis model, along with different variations developed by other scholars, has played a crucial role
in advancing mathematical research. Much of this research has focused on whether the solutions to these
models exhibit blow-up behavior or not (see, e.g., [1,12,14,32,38,40]).
The biological significance of the flux limitation in the cross-diffusion term derives from the fact that
living organisms can sense not only density but also gradient. As a refinement of the classical Keller–Segel
system, this particular feature can be introduced in the general chemotactic system:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
ut=Δu−div(χuf (|∇v|)·∇v),x∈Ω,t>0,
0=Δv−M+u, x ∈Ω,t>0,
∂νu=∂νv=0,x∈∂Ω,t>0,
u(x, 0) = u0(x),x∈Ω,
(1.1)
where M=1
|Ω|
Ω
u0(x)dx. In the case where f(|∇v|)=|∇v|p−2, Negreanu and Tello [22] established the
existence of globally bounded solutions when p>1andn=1or1<p< n
n−1and n≥2, and of infinitely
many non-constant steady states, provided that 1 <p<2andn= 1, whereas the blow-up of solutions
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