TY - JOUR

T1 - Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems

AU - Black, Tobias

AU - Fuest, Mario

AU - Lankeit, Johannes

N1 - Funding Information: Open Access funding enabled and organized by Projekt DEAL. The second author is partly supported by the German Academic Scholarship Foundation and by the Deutsche Forschungsgemeinschaft within the project Emergence of structures and advantages in cross-diffusion systems, project number 411007140.

PY - 2021/4/30

Y1 - 2021/4/30

N2 - We study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In \(n\)-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 = \Delta v - \frac1{|\Omega|} \int_\Omega u + u \end{cases} \tag{JL} \end{align*} and \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 = \Delta v - v + u, \end{cases}\tag{PE} \end{align*} where \(\lambda\) and \(\mu\) are given spatially radial nonnegative functions and \(m, \kappa > 0\) are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for \(m,\kappa\) leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant \(\lambda, \mu > 0\), we find that there are initial data which lead to blow-up in (JL) if \begin{alignat*}{2} 0 \leq \kappa &< \min\left\{\frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&&\qquad\text{if } m\in\left[\frac{2}{n},\frac{2n-2}{n}\right)\\ \text{ or }\quad 0 \leq \kappa&

AB - We study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In \(n\)-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 = \Delta v - \frac1{|\Omega|} \int_\Omega u + u \end{cases} \tag{JL} \end{align*} and \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 = \Delta v - v + u, \end{cases}\tag{PE} \end{align*} where \(\lambda\) and \(\mu\) are given spatially radial nonnegative functions and \(m, \kappa > 0\) are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for \(m,\kappa\) leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant \(\lambda, \mu > 0\), we find that there are initial data which lead to blow-up in (JL) if \begin{alignat*}{2} 0 \leq \kappa &< \min\left\{\frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&&\qquad\text{if } m\in\left[\frac{2}{n},\frac{2n-2}{n}\right)\\ \text{ or }\quad 0 \leq \kappa&

KW - math.AP

KW - 35B44 (primary), 35K55, 92C17 (secondary)

KW - Logistic source

KW - Finite-time blow-up

KW - Nonlinear diffusion

KW - Chemotaxis

UR - http://www.scopus.com/inward/record.url?scp=85105199302&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2005.12089

DO - 10.48550/arXiv.2005.12089

M3 - Article

VL - 72

JO - Zeitschrift fur Angewandte Mathematik und Physik

JF - Zeitschrift fur Angewandte Mathematik und Physik

SN - 0044-2275

IS - 3

M1 - 96

ER -