Blow‐up phenomena in chemotaxis systems with a source term

Abstract

This paper dealswith a parabolic–parabolic Keller–Segel-type systemin a bounded domain ofRN, fN D 2; 3g, under different\ud boundary conditions, with time-dependent coefficients and a positive source term. The solutions may blowup in finite\ud time t; and under appropriate assumptions on data, explicit lower bounds for blow-up time are obtained when blow up\ud occur

Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth

“…Furthermore, a number of interesting results concerning properties of solutions to chemotaxis-systems have been also attained for a broader class of problems, in which the first equation of (1) reads u t = ∇ • (S(u)∇u) − ∇ • (T (u)∇v). Precisely, bounded or unbounded solutions of the corresponding problem is determined by the asymptotic behaviour of the ratio T (u)/S(u), especially in terms of the space dimension; we refer, for instance, to [6] and [40] for the parabolic-elliptic case and to [9,17,27,28,38] for the parabolic-parabolic case.…”

“…which implies (14a) and (14b). Hence, (14c) results from (17) and (14a) and the non negativity of Ω u ε . Now, for any t 2 > t 1 , let us independently estimate the terms…”

Effect of nonlinear diffusion on a lower bound for the blow-up time in a fully parabolic chemotaxis system

“…This restriction on m 2 and n was removed by Anderson-Deng [1] when Ω is a convex bounded domain and m 1 = 1. Furthermore, as a new attempt to estimating a lower bound for the blow-up time in the above sense, Marras-Vernier Piro-Viglialoro [21,22] obtained a lower bound for the blow-up time of the more generalized equation with a source term:…”

Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity, and logistic source

“…Additionally, for the same problem (5), also defined in a convex smooth and bounded domain Ω of R n , n ≥ 1, but with source term g such that −c 0 (s + s α ) ≤ g(s) ≤ a − bs α , for s ≥ 0, and with some α > 1, a ≥ 0 and χ, b, c 0 > 0, global existence of very weak solutions, as well their boundedness properties and long time behavior are discussed in [26], [27] and also [28]. Finally, for the sake of completeness, it is worth to precise that even though in the logistic source the term −µu 2 , with µ > 0, corresponds to a death rate of the cells distribution and generally contrasts blow-up phenomena, in [31] is shown that under radially symmetric assumptions there exist initial data such that the corresponding solution of systems type (5) blows up (we also refer to [14] for techniques used to estimate the blow up time of unbounded solutions to system related to (5)).…”