Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system

This work considers the Keller-Segel-type parabolic system {u(t) = Delta(u phi(v)), x is an element of Omega, t > 0, v(t) = Delta v - v + u, x is an element of Omega, t > 0, ( *) in a smoothly bounded convex domain Omega subset of R-n, n >= 2, under no-flux boundary conditions, which has recently been proposed as a model for processes of stripe pattern formation via so-called self-trapping mechanisms. In the two-dimensional case, in stark contrast to the classical Keller-Segel model in which large-data solutions may blow up in finite time, for all suitably regular initial data the associated initial value problem is seen to possess a globally-defined bounded classical solution, provided that the motility function phi is an element of C-3 ([0, infinity)) boolean AND W-1,W-infinity ((0,infinity)) is uniformly positive. In the corresponding higher-dimensional setting, it is shown that certain weak solutions exist globally, where in the particular three-dimensional case this solution actually is bounded and classical if the initial data are suitably small in the norm of L-2 (Omega) x W-1,W-4 (Omega). Finally, if still n = 3 but merely the physically interpretable quantity parallel to phi 'parallel to(L)infinity (((0, infinity))) integral(Omega) u(0) is appropriately small, then the above-weak solutions are proved to become eventually smooth and bounded.