The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties

Abstract

We consider the chemotaxis system [Formula: see text] as originally introduced in 1971 by Keller and Segel in the second of their seminal works. This system constitutes a prototypical model for taxis-driven pattern formation and front propagation in various biological contexts such as tumor angiogenesis, but in the higher-dimensional context any global existence theory for large-data solutions is yet lacking. In this work it is shown that in bounded planar domains [Formula: see text] with smooth boundary, for all reasonably regular initial data [Formula: see text] and [Formula: see text], the corresponding Neumann initial-boundary value problem possesses a global generalized solution. Thus particularly addressing arbitrarily large initial data, this goes beyond previously gained results asserting global existence of solutions only in spatial one-dimensional problems, or under certain smallness conditions on the initial data. The derivation of this result is based on a priori estimates for the quantities [Formula: see text] and [Formula: see text] in spatio-temporal [Formula: see text] spaces, where further boundedness and compactness properties are derived from the former by relying on the planar spatial setting in using an associated Moser–Trudinger inequality. Furthermore, some further boundedness and relaxation properties are derived, inter alia indicating that for any such solution we have [Formula: see text] in [Formula: see text] as [Formula: see text] for all finite [Formula: see text], and that in an appropriate generalized sense the quantities [Formula: see text] and [Formula: see text] eventually enter bounded sets in [Formula: see text] and [Formula: see text], respectively, with diameters only determined by the total population size [Formula: see text]. Finally, some numerical experiments illustrate the analytically obtained results.