Application of an Adams type inequality to a two-chemical substances chemotaxis system

This paper deals with positive solutions of the fully parabolic system, {ut = Delta u - chi del center dot (u del v) in Omega x (0, infinity), tau(1)v(t) = Delta v - v + w in Omega x (0, infinity), tau(2)w(t) = Delta w - w + u in Omega x (0, infinity), under homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded domain Omega subset of R-n (n <= 4) with positive parameters tau(1), tau(2), chi > 0 and nonnegative smooth initial data (u(0), v(0), w(0)). In the lower dimensional case (n <= 3), it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded. In the case n = 4, it is shown that in the radially symmetric setting solutions to the Neumann boundary value problem of the system exist globally in time and remain bounded if vertical bar vertical bar u(0)vertical bar vertical bar(L1(Omega)) < (8 pi)(2)/chi; as to the mixed boundary value problem, we will establish global existence and boundedness of solutions if vertical bar vertical bar u(0)vertical bar vertical bar(L1(Omega)) < (8 pi)(2)/chi without radial symmetry. The key ingredients are a Lyapunov functional and an Adams type inequality. A Lyapunov functional of the above problems will be constructed and the constant (8 pi)(2)/chi is deduced from the critical constant in the Adams type inequality. This result is regarded as a generalization of the well-known 8 pi problem in the Keller-Segel system to higher dimensions. (C) 2017 Elsevier Inc. All rights reserved.