Blow-up profiles and refined extensibility criteria in quasilinear Keller-Segel systems

In this work we consider the system{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)inΩ×(0,∞)vt=Δv−v+uinΩ×(0,∞), for a bounded domain Ω⊂Rn, n≥2, where the functions D and S behave similarly to power functions. We prove the existence of classical solutions under Neumann boundary conditions and for smooth initial data. Moreover, we characterise the maximum existence time Tmax of such a solution depending chiefly on the relation between the functions D and S: We show that a finite maximum existence time also results in unboundedness in Lp-spaces for smaller p∈[1,∞).