Finite-time blow-up to hyperbolic Keller-Segel system of consumption type with logarithmic sensitivity

Abstract

This paper deals with a hyperbolic Keller-Segel system of consumption type with the logarithmic sensitivity \begin{equation*} \partial_{t} \rho = - \chi\nabla \cdot \left (\rho \nabla \log c\right),\quad \partial_{t} c = - \mu c\rho\quad (\chi,\,\mu>0) \end{equation*} in $\mathbb{R}^d\; (d \ge1)$ for nonvanishing initial data. This system is closely related to tumor angiogenesis, an important example of chemotaxis. We firstly show the local existence of smooth solutions corresponding to nonvanishing smooth initial data. Next, through Riemann invariants, we present some sufficient conditions of this initial data for finite-time singularity formation when $d=1$. We then prove that for any $d\ge1$, some nonvanishing $C^\infty$-data can become singular in finite time. Moreover, we derive detailed information about the behaviors of solutions when the singularity occurs. In particular, this information tells that singularity formation from some initial data is not because $c$ touches zero (which makes $\log c$ diverge) but due to the blowup of $C^1\times C^2$-norm of $(\rho,c)$. As a corollary, we also construct initial data near any constant equilibrium state which blows up in finite time for any $d\ge1$. Our results are the extension of finite-time blow-up results in \cite{IJ21}, where initial data is required to satisfy some vanishing conditions. Furthermore, we interpret our results in a way that some kinds of damping or dissipation of $\rho$ are necessarily required to ensure the global existence of smooth solutions even though initial data are small perturbations around constant equilibrium states.