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

Abstract

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) + λu - μu^{1+κ}, \\ 0 = Δv - \frac1{|Ω|} \int_Ωu + 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) + λu - μu^{1+κ}, \\ 0 = Δv - 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 κ&< \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 κ&<\min\left\{\frac{1}{2},\frac{n-1}n-\frac{m}2\right\} &&\qquad \text{if } m\in\left(0,\frac{2}{n}\right), \end{alignat*} and in (PE) if $m\in [1,\frac{2n-2}{n})$ and \begin{align*} 0 \leq κ< \min\left\{\frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\}. \end{align*}

21 pages

References in corpus (1)

• New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models

Cited by in corpus (6)

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• Strong convergence of weighted gradients in parabolic equations and applications to global generalized solvability of cross-diffusive systems
• Can chemotactic effects lead to blow-up or not in two-species chemotaxis-competition models?
• Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source