Global Boundedness of the Fully Parabolic Keller-Segel System with Signal-Dependent Motilities

This paper establishes the global uniform-in-time boundedness of solutions to the following Keller-Segel system with signal-dependent diffusion and chemotaxis {u(t) = del.(gamma(v)del u - u phi(v)del v), x is an element of Omega, t > 0, v(t) = d Delta v - v + u, x is an element of Omega, t > 0 in a bounded domain Omega subset of R-N(N <= 4) with smooth boundary, where the density-dependent motility functions gamma(v) and phi(v) denote the diffusive and chemotactic coefficients, respectively. The model was originally proposed by Keller and Segel in (J. Theor. Biol. 30:225-234, 1970) to describe the aggregation phase of Dictyostelium discoideum cells, where the two motility functions satisfy a proportional relation phi(v) = (alpha - 1)gamma '(v) with alpha > 0 denoting the ratio of effective body length (i.e. distance between receptors) to the step size. The major technical difficulty in the analysis is the possible degeneracy of diffusion. In this work, we show that if gamma(v) > 0 and phi(v) > 0 are smooth on [0, infinity) and satisfy inf(v >= 0) d gamma(v)/v phi(v)(v phi(v) + d - gamma(v))(+) > N/2, then the above Keller-Segel system subject to Neumann boundary conditions admits classical solutions uniformly bounded in time. The main idea of proving our results is the estimates of a weighted functional integral(Omega) u(p)v(-q)dx for p > N/2 by choosing a suitable exponent p depending on the unknown v, by which we are able to derive a uniform L-infinity-norm of v and hence rule out the diffusion degeneracy.

Automated summary

This paper establishes the global uniform-in-time boundedness of solutions to the Keller-Segel system with signal-dependent diffusion and chemotaxis. The analysis involves estimating a weighted functional integral to derive a uniform L-infinity-norm of the solution and rule out the degeneracy of diffusion.