A refined criterion and lower bounds for the blow-up time in a parabolic–elliptic chemotaxis system with nonlinear diffusion. (June 2020) - Marras, Monica; Nishino, Teruto; Viglialoro, Giuseppe
Abstract: This paper deals with unbounded solutions to the following zero-flux chemotaxis system ( ♢ ) u t = ∇ ⋅ [ ( u + α ) m 1 − 1 ∇ u − χ u ( u + α ) m 2 − 2 ∇ v ] ( x, t ) ∈ Ω × ( 0, T m a x ), 0 = Δ v − M + u ( x, t ) ∈ Ω × ( 0, T m a x ), where α > 0, Ω is a smooth and bounded domain of R n, with n ≥ 1, t ∈ ( 0, T m a x ) is T m a x the blow-up time, and m 1, m 2 are real numbers. Given a sufficiently smooth initial data u 0 ≔ u ( x, 0 ) ≥ 0 and set M ≔ 1 | Ω | ∫ Ω u 0 ( x ) d x, from the literature it is known that under a proper interplay between the above parameters m 1, m 2 and the extra condition ∫ Ω v ( x, t ) d x = 0, system ( ♢ ) possesses for any χ > 0 a unique classical solution which becomes unbounded at t ↗ T m a x . In this investigation we first show that for p 0 > n 2 ( m 2 − m 1 ) 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 T m a x providing a lower bound T .