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Finite time collapse in chemotaxis systems with logistic‐type superlinear source
Author(s) -
Marras Monica,
VernierPiro Stella
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6676
Subject(s) - mathematics , homogeneous , ball (mathematics) , neumann boundary condition , norm (philosophy) , upper and lower bounds , type (biology) , combinatorics , function (biology) , mathematical analysis , boundary value problem , evolutionary biology , political science , law , biology , ecology
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.