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 .

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system  under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in R n with smooth boundary, α > 0, χ > 0, m 1 , m 2 ∈ R and T > 0. Recently, Anderson-Deng [1] gave a lower bound for the blow-up time in the case that m 1 = 1 and Ω is a convex bounded domain. The purpose of this paper is to generalize the result in [1] to the case that m 1 = 1 and Ω is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo-Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of Ω.2010Mathematics Subject Classification. Primary: 35B44, 35K51, 35K59; Secondary: 35Q92, 92C17.

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