Global boundedness and finite time blow-up of solutions for a quasilinear chemotaxis-May-Nowak model

TL;DR

The study analyzes a quasilinear chemotaxis-May-Nowak model with nonlinear diffusion $D(u)$ that generalizes $(1+u)^{m-1}$, examining how diffusion strength interacts with chemotaxis to determine global existence and blow-up. Using local well-posedness results and maximal Sobolev regularity, it derives a priori estimates that lead to global boundedness under critical diffusion exponents for both the parabolic-elliptic-parabolic ($\kappa=0$) and fully parabolic ($\kappa=1$) cases, while showing finite-time blow-up in low dimensions for weak diffusion ($m<1$) under radially symmetric data. The results specify precise thresholds on $m$ (depending on the spatial dimension) that separate boundedness from blow-up, enhancing understanding of diffusion-chemotaxis balance in spatial viral dynamics models. Overall, the work provides rigorous criteria for when nonlinear diffusion prevents aggregation and when it fails to avert blow-up, with methodological emphasis on maximal Sobolev regularity and iterative $L^p$-bounds.

Abstract

In this paper, we introduce the nonlinear diffusion term $\nabla\cdot(D(u)\nabla u)$ into the chemotaxis-May-Nowak model to investigate the effects of $D(u)$ and chemotaxis on the global existence, boundedness, and finite time blow-up of solutions. Here, $D(u)$ generalizes the prototype $(1+u)^{m-1}$ with $m\in\R$. For the parabolic-elliptic-parabolic case, if $m>2+\frac{n}{2}-\frac2{n}$ when $n\ge3$ and $m>\frac32$ when $n=2$, then all solutions exist globally and remain bounded, whereas if $n\in\{2,3\}$ and $m<1$, finite time blow-up occurs when $Ω$ is a ball and the initial data are radially symmetric. For the fully parabolic case, if $m>1+\frac{n}{2}-\frac2{n}$, then all solutions exist globally and remain bounded.

Theorems & Definitions (21)

• Lemma 1.1
• Theorem 1.1
• Theorem 1.2
• Proposition 1.1
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5