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

We consider the following quasilinear Keller–Segel systemu t = Δ u − ∇ ( u ∇ v ) + g ( u ) ,( x , t ) ∈ Ω × [ 0 , T m a x ) ,0 = Δ v − v + u ,( x , t ) ∈ Ω × [ 0 , T m a x ) ,on a ball Ω ≡ B R ( 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 )= λ u − μ u k , k >1, μ >0, λ >0, and T m a x 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 L p (Ω)‐norm for some p >1. Moreover, a lower bound of the lifespan (or blow‐up time when it is finite) T m a x is derived. In addition, if Ω ⊂ ℝ 3a lower bound of T m a x is explicitly computable.