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

We consider the chemotaxis system {u(t) = Delta u - del . (u/v del u), v(t) = Delta v - uv, 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 Omega with smooth boundary, for all reasonably regular initial data u(0) >= 0 and v(0) > 0, 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 del ln(u + 1) and del(v) in spatio-temporal L-2 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 v(., t) -> 0 in L-p(Omega) as t -> infinity for all finite p > 1, and that in an appropriate generalized sense the quantities u and del ln v eventually enter bounded sets in Lp(O) and L2(O), respectively, with diameters only determined by the total population size integral(Omega) u(0). Finally, some numerical experiments illustrate the analytically obtained results.