Blowup in infinite time of radial solutions for a parabolic-elliptic system in high-dimensional Euclidean spaces
type: Item
instance of: scholarly article
title: Blowup in infinite time of radial solutions for a parabolic-elliptic system in high-dimensional Euclidean spaces
zbMATH Open document ID: 1172.35003
author: Takasi Senba
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-03-25T00:00:00Z
review text: Given two integers \(N\geq 11\) and \(J\geq 0\), define \[ \nu=\frac{-N-2 + \sqrt{(N-10)(N-2)}}{4} \;\;\;\text{ and }\;\;\; a_J = \frac{N+2J+6\nu+4}{-2\nu-2}\;. \] A global radially symmetric solution \((U,V)\) to the Smoluchowski-Poisson system \[ \partial_t U = \nabla\cdot\left( \nabla U - U \nabla V \right)\;, \qquad -\Delta V = U \;\;\;\text{ in }\;\;\; (0,\infty)\times{\mathbb R}^N \] is constructed with the following properties: \[ \sup_{(x,t)\in {\mathbb R}^N\times (0,\infty)}{\left\{ |x|^2 u(x,t) \right\}}<\infty\;, \qquad c t^{a_J} \leq \|U(t)\|_\infty \leq C t^{a_J} \] for \(t\geq T\), for some constants \(C>c>0\) and \(T>0\). Furthermore, if \(U_0\) is a radially symmetric, nonnegative, and continuous initial condition satisfying \(U_0(x)\leq 2(N-2)/|x|^2\) for \(x\in {\mathbb R}^N\), the corresponding solution \((U,V)\) to the Smoluchowski-Poisson system exists globally with \(\|U(t)\|_\infty \leq C t^{a_0}\) for \(t\) large enough. Similar results were already obtained for \(\partial_t u = \Delta u + u^p\) in [\textit{N.~Mizoguchi}, J. Differ. Equations 227, 652--669 (2006; Zbl 1102.35049)] and the same strategy of proof is used here.
reviewed by: Philippe Laurençot
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 5532901
zbMATH Keywords: unbounded global solutions
zbMATH Keywords: temporal growth estimates
zbMATH Keywords: Keller-Segel system
zbMATH Keywords: growup
zbMATH Keywords: Smoluchowski-Poisson system
MaRDI profile type: Publication
full work available at URL: j.na.2008.03.041
OpenAlex ID: W1983298524
cites work: Q5554503
cites work: Existence and nonexistence of solutions for a model of gravitational interaction of particles, II
cites work: The 8π‐problem for radially symmetric solutions of a chemotaxis model in the plane
cites work: Optimal critical mass in the two dimensional Keller-Segel model in \(R^2\)
cites work: Self-similar blow-up for a reaction-diffusion system
cites work: Growup of solutions for a semilinear heat equation with supercritical nonlinearity
cites work: Q5446138
cites work: Q5446136
cites work: Blowup Behavior of Radial Solutions to Jaeger-Luckhaus System in High Dimensional Domains
cites work: A fast blowup solution to an elliptic-parabolic system related to chemotaxis
DOI: 10.1016/J.NA.2008.03.041
Recommended article: Degenerate parabolic equation with critical exponent derived from the kinetic theory. III: \(\varepsilon \)-regularity.
Recommended article: Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system
Recommended article: Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system
Recommended article: Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species.
Recommended article: Global Existence vs. Blowup in a One-dimensional Smoluchowski-Poisson System
Recommended article: Blowup in infinite time for 2D Smoluchowski-Poisson equation.
Recommended article: Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models
Recommended article: Almost collapse mass quantization in 2D Smoluchowski-Poisson equation
Recommended article: Volume effects in the Keller--Segel model: energy estimates preventing blow-up
Recommended article: Q4712903
description: scientific article; zbMATH DE number 5532901