A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity

We consider the parabolic chemotaxis model {u(t) = Delta u - chi del .(u/v del v) v(t) = Delta v - v + u in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for chi is an element of E (0, chi(0)) for some chi(0) > 1, thereby proving that the value chi = 1 is not critical in this regard. Our main tool is consideration of the energy functional F-a,F-b(u, v) = integral(Omega) u ln u - a integral(Omega) u ln u + b integral(Omega) vertical bar del root v vertical bar(2) for a > 0, b >= 0, where using nonzero values of b appears to be new in this context. Copyright 2016 John Wiley & Sons, Ltd.