A Blow-Up Result for the Chemotaxis System with Nonlinear Signal Production and Logistic Source

In this paper we consider the following chemotaxis-growth system with nonlinear signal production and logistic source {u(t) = Delta u - chi del . (u del v) + lambda u - mu u(alpha), x is an element of Omega, t > 0, 0 = Delta v - mu(t) + f(u), mu(t) :=1 vertical bar Omega vertical bar integral(Omega) f(u(., t)), x is an element of Omega, t .0, with homogeneous Neumann boundary conditions in the ball Omega = B-R(0) subset of R-n for n >= 1 and R > 0, where chi, lambda, mu > 0, alpha > 1, and f is an appropriate regular function satisfying f (u) >= ku(kappa) for all u >= 1, kappa > 0 with some constants k > 0. If the number kappa and alpha satisfy kappa + 1 > alpha (2/n +1). then there exists appropriate initial data such that the corresponding solution (u, v) of the system blow up in finite time. This result extends the blow-up result of the chemotaxis model without logistic cell kinetics in [45]. Apparently, for the case kappa = 1, this provides a rigorous detection for blow-up of solution in spaces-dimensions three and four.