On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis (Q2782953)

The authors consider a system of a parabolic equation and an o.d.e. introduced by \textit{H. Othmer} and \textit{A. Stevens} [SIAM J. Appl. Math. 57, 1044-1081 (1997; Zbl 0990.35128)] that serves as a odel for chemotaxis: \(p_t=D\nabla \cdot(p\nabla \ln(p/w))\) and \(w_t=F(p,w)\) in \(\Omega\) with boundary condition \(\vec n\cdot p\nabla \ln=0\) on \(\partial\Omega \times \mathbb{R}^+\) and initial conditions \(p(x,0)= p_0(x)\) and \(w(x,0)= w_0(x)\). They extend results of \textit{H. A. Levine} and \textit{B. D. Sleeman} in [SIAM J. Appl. Math. 57, 683-730 (1997; Zbl 0874.35047)] for the system above assuming that the o.d.e is as \(w_t=\beta p-\mu w\), or \(w_t=(\beta p-\mu)w\). A large part of the paper is concerned with explicit estimates for the case of one space variable.