A refined criterion and lower bounds for the blow-up time in a parabolic–elliptic chemotaxis system with nonlinear diffusion

Abstract

This paper deals with unbounded solutions to the following zero-flux chemotaxis systemwhere α > 0, Ω is a smooth and bounded domain of R n , with n ≥ 1, t ∈ (0, Tmax), where Tmax the blow-up time, and m1, m2 real numbers. Given a sufficiently smooth initial data u0 := u(x, 0) ≥ 0 and set M := 1 |Ω| Ω u0(x) dx, from the literature it is known that under a proper interplay between the above parameters m1, m2 and the extra condition Ω v(x, t)dx = 0, system ( ) possesses for any χ > 0 a unique classical solution which becomes unbounded at t ր Tmax. In this investigation we first show that for p0 > n 2 (m2 − m1) any blowing up classical solution in L ∞ (Ω)-norm blows up also in L p 0 (Ω)-norm. Then we estimate the blow-up time Tmax providing a lower bound T .

Boundedness in a three‐dimensional two‐species and two‐stimuli chemotaxis system with chemical signalling loop

“…where we used the fact | v| 2 ≤ n|D 2 v| 2 . Using Young's inequality to the first and the second terms of right-hand side of (23), the treatments are similar to (18) and (19), reorganizing the terms then we obtain (21). Similarly, we derive (22).…”

“…16 For more results about one-specie and one-stimuli chemotaxis system, we refer the readers to previous studies. [17][18][19][20][21][22][23][24] The following is the two-species and two-stimuli chemotaxis system under Lotka-Volterra competitive kinetics:…”

“…However, for parabolic‐elliptic simplifications of (), the logistic dampening cannot completely exclude the possibility of blow‐up in n ≥ 3 15 and n ≥ 5 16 . For more results about one‐specie and one‐stimuli chemotaxis system, we refer the readers to previous studies 17‐24 …”

Finite time collapse in chemotaxis systems with logistic‐type superlinear source

“…from which we obtain (17): it means that the solution of (1) exists bounded in the interval [0, T], with T = 1 2AΦ 2…”

“…under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in R n with smooth boundary, > 0, > 0, m 1 , m 2 ∈ R, and T > 0, Nishino and Yokota 16 derived a lower bound of blow-up time. • If = 0, > 0, g(u) = 0, and M ∶= 1 |Ω| ∫ Ω u 0 (x) dx, Marras et al 17 investigate the blow-up solutions of the following:…”