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

We consider the following quasilinear Keller–Segel system u_t=Δu−∇(u∇v)+g(u), (x,t)∈Ω×[0,T_max)), 0=Δv−v+u, (x,t)∈Ω×[0,T_max), on a ball Ω≡_(0)⊂ℝ^n n ≥ 3, R>0, under homogeneous Neumann boundary conditions and nonnegative initial data. The source term g(u) is superlinear and of logistic type, that is, g(u)=λu−μuk,k>1,μ>0, λ>0, and Tmax is the blow‐up time. The solution (u,v) may or may not blow‐up in finite time. Under suitable conditions on data, we prove that the function u, which blows up in L∞(Ω)‐norm, blows up also in Lp(Ω)‐norm for some p>1. Moreover, a lower bound of the lifespan (or blow‐up time when it is finite) Tmax is derived. In addition, if Ω⊂ℝ^3 a lower bound of Tmax is explicitly computable.