Ultrasonic inspection devices : Dr J.U.H. Krautkramer Ges fur Elektrophysik, British Patent 1,224,821 (1971) Filed 6 March 1969

Abstract

The quasilinear chemotaxis system is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, n≥2, with smooth boundary, where the focus is on cases when herein the diffusivity D(s) decays exponentially as s→∞.It is shown that under the subcriticality condition that with some C>0 and α<2n, for all suitably regular initial data satisfying an essentially explicit smallness assumption on the total mass ∫Ωu0, the corresponding Neumann initial–boundary value problem for (⋆) possesses a globally defined bounded classical solution which moreover approaches a spatially homogeneous steady state in the large time limit. Viewed as a complement of known results on the existence of small-mass blow-up solutions in cases when in (0.1) the reverse inequality holds with some α>2n, this confirms criticality of the exponent α=2n in (0.1) with regard to the singularity formation also for arbitrary n≥2, thereby generalizing a recent result on unconditional global boundedness in the two-dimensional situation.As a by-product of our analysis, without any restriction on the initial data, we obtain boundedness and stabilization of solutions to a so-called volume-filling chemotaxis system involving jump probability functions which decay at sufficiently large exponential rates.