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

Abstract

The chemotaxis system(⋆){ut=∇⋅(D(u,v)∇u)−∇⋅(S(u,v)∇v),vt=Δv−v+u, is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, 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−1andS(u,v)=u(u+1)σ−1,u≥0,v≥0, with m∈R and σ∈R such that(⋆⋆⋆)m−n−2n<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.