Boundedness vs. blow-up in a chemotaxis system

Abstract

We determine the critical blow-up exponent for a Keller-Segel-type chemotaxis model, where the chemotactic sensitivity equals some nonlinear function of the particle density. Assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution. Furthermore, we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails.

<i>L</i>_<i>p</i>-Estimates of Solutions to <i>n</i>-Dimensional Parabolic-Parabolic System for Chemotaxis with Subquadratic Degradation

“…The existence interval can be again extended, to ½0; T þ 2T T . Iteration of this procedure proves the global existence theorem with the estimate (5). r…”

Boundedness of the solution of a higher-dimensional parabolic–ODE–parabolic chemotaxis–haptotaxis model with generalized logistic source

“…To begin with, let us collect some basic solution properties which essentially have already been used in [10] (see also Winkler [38], Zhang and Li [42]). …”

“…In 1970s, a well-known chemotaxis model was proposed by Keller and Segel ([13]), which describes the aggregation processes of the cellular slime mold Dictyostelium discoideum. Since then, a number of variations of the Keller-Segel model have attracted the attention of many mathematicians, and the focused issue was the boundedness or blow-up of the solutions ( [5,7,9,10,39,20]). The striking feature of Keller-Segel models is the possibility of blow-up of solutions in a finite (or infinite) time (see, e.g., [1,9,18,39]), which strongly depends on the space dimension.…”

“…Then we shall involve the variation-of-constants formula for variable v to gain Ω |∇v| q ≤ C for all and q ∈ [1, 10) and thereby establish the a priori estimates of the functional …”

Global Existence and Blow-Up for a Chemotaxis System

“…Dynamics and sensitivity of the solutions for a bacterial self-organization model is investigated in [4]. Since the existence of cell kinetics term f (u, w) and without the assumption of radially symmetric in this paper, we can't construct effective energy function as done in [5], [11]. Our future research is to study the blow-up issue of the system (1.1).…”