Global classical solvability and generic infinite-time blow-up in quasilinear Keller–Segel systems with bounded sensitivities

By Michael Winkler. Journal of Differential Equations, 2019

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Abstract The chemotaxis system (⋆) { u t = ∇ ⋅ ( D ( u , v ) ∇ u ) − ∇ ⋅ ( S ( u , v ) ∇ v ) , v t = Δ v − v + u , is considered under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 2 , along with initial conditions involving suitably regular and nonnegative data. It is firstly asserted that if the positive smooth function D decays at most algebraically with respect to u, then for any smooth nonnegative and bounded S fulfilling a further mild assumption especially satisfied when S ≡ S ( u ) with S ( 0 ) = 0 , (⋆) possesses a globally defined classical solution. If Ω is a ball, then under appropriate assumptions on D and S generalizing the prototypical choices in (⋆⋆) D ( u , v ) = ( u + 1 ) m − 1 and S ( u , v ) = u ( u + 1 ) σ − 1 , u ≥ 0 , v ≥ 0 , with m ∈ R and σ ∈ R such that (⋆⋆⋆) m − n − 2 n σ ≤ 0 , the phenomenon of infinite-time blow-up is next shown to occur for all initial data within a set B of functions which inter alia is found to be dense in the set of all radially symmetric and suitably regular positive functions on Ω ‾ . Up to equality in (⋆⋆⋆) thereby covering the largest possible range of nonpositive σ for the appearance of unbounded solutions, this extends previous findings on blow-up in infinite time which in the context of (⋆⋆) were limited to a smaller parameter region, and which were restricted to mere existence results without information on the richness of B .

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

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