Global existence and slow grow-up in a quasilinear Keller–Segel system with exponentially decaying diffusivity

Abstract

The Neumann initial-boundary value problem for the chemotaxis system{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),vt=Δv−v+u,(⋆)is considered in a bounded domain , , with smooth boundary. In compliance with refined modeling approaches, the diffusivity function D therein is allowed to decay considerably fast at large densities, where a particular focus will be on the mathematically delicate case when D(s) decays exponentially as . In such situations, namely, straightforward Moser-type recursive arguments for the derivation of estimates for u from corresponding Lp bounds seem to fail. Accordingly, results on global existence, and especially on quantitative upper bounds for solutions, so far mainly concentrate on cases when D decays at most algebraically, and hence are unavailable in the present context.This work develops an alternative approach, at its core based on a Moser-type iteration for the quantity , to establish global existence of classical solutions for all reasonably regular initial data, as well as a logarithmic upper estimate for the possible growth of as , under the assumptions that with some K1  >  0, K2  >  0, and we have for all , and that the size of S relative to D can be estimated according to for all with some K3  >  0 and .Making use of the fact that this allows for certain superalgebraic growth of , as a particular consequence of this and known results on nonexistence of global bounded solutions we shall see that in the prototypical case when and for all and some positive α and β, the assumptions that and that0\\quad \\quad \ ext{and}\\quad \\quad \\left\\{\\begin{array}{*{35}{l}} \\alpha \\in \\left(\\frac{\\beta}{2},\\beta \\right)\\quad \\quad & \ ext{if}~n=2, \\\\ \\alpha \\in \\left(\\frac{\\beta}{2},\\beta \\right]\\quad \\quad & \ ext{if}~n\\geqslant 3, \\end{array}\\right. \\end{eqnarray} ?>β>0and{α∈(β2,β)if n=2,α∈(β2,β]if n⩾3,warrant the existence of classical solutions which are global but unbounded, and for which this infinite-time blow-up is slow in the sense that the corresponding grow-up rate is at most logarithmic.To the best of our knowledge, this inter alia seems to constitute the first quantitative information on a blow-up rate in a parabolic Keller–Segel system of type () for widely arbitrary initial data, hence independent of a particular construction of possibly non-generic exploding solutions.