Optimal mass on the parabolic-elliptic-ODE minimal chemotaxis-haptotaxis in R2

Abstract

In this paper, we study the Cauchy problem to a chemotaxis-haptotaxis model describing cancer invasion in $\mathbb{R}^{2}$. The main feature is to prove that $8\pi$ is the critical mass on initial data for distinguishing existence and blow-up of solutions to the model. Namely, when the initial mass is less than $8\pi$, we prove global existence of solutions by constructing a proper free energy and using the Brezis-Merle type inequality. On the contrary, the finite time blow-up of solutions may occur if the initial mass is greater than $8\pi$ and the initial second moment is small enough. Comparing to the classical Keller-Segel model, we find that the haptotaxis has no effect on the critical mass of the model.