Behavior in time of solutions to a degenerate chemotaxis system with flux limitation
Monica Marras, S. Vernier-Piro, Tomomi Yokota
Abstract
We study a new class of Keller–Segel models, which presents a limited flux and an optimal transport of cells density according to chemical signal density. As a prototype of this class we study radially symmetric solutions to the parabolic–elliptic system u t = ∇ ⋅ ( u ∇ u u 2 + | ∇ u | 2 ) − χ k f ∇ ⋅ ( u ∇ v ( 1 + | ∇ v | 2 ) α ) , x ∈ Ω , t > 0 , 0 = Δ v − μ + u , x ∈ Ω , t > 0 under no flux boundary conditions in a ball B = Ω ⊂ R N and initial condition u ( x , 0 ) = u 0 ( x ) > 0 , χ > 0 , α > 0 , k f > 0 and μ = 1 | Ω | ∫ Ω u 0 d x . Under suitable conditions on α and u 0 it is shown that the solution blows up in L ∞ -norm at a finite time T m a x and for some p > 1 it blows up also in L p -norm. The proofs are mainly based on an helpful change of variables, on comparison arguments and some suitable estimates.