On a quasilinear fully parabolic attraction or repulsion chemotaxis system with nonlinear signal production
The existence of non-negative global bounded classical solutions to the chemotaxis system \begin{align*} \partial_t u & = \mathrm{div}\big( \phi(u) \nabla u - \chi \psi(u)\nabla v \big)\text{ in } (0,\infty)\times\Omega, \\ \partial_t v & = \Delta v - v + g(u)\text{ in } (0,\infty)\times\Omega, \end{align*} supplemented with no-flux boundary conditions and non-negative initial conditions \((u_0,v_0)\in C(\bar{\Omega})\times C^1(\bar{\Omega})\), is investigated when the chemotactic sensitivity \(\chi\) is either positive (chemo-attraction) or negative (chemo-repulsion) and the \(C^2\)-smooth nonlinearities \(\phi\), \(\psi\), and \(g\) satisfy \begin{align*} & K_0 (s+1)^m \le \phi(s), \\ & 0 < \psi(s)\le K_1 (s+1)^\alpha, \quad \psi(0)=0,\\ & 0 < g(s) \le K_2 (s+1)^\beta, \end{align*} for \(s>0\) and some constants \((K_0,K_1,K_2,\beta)\in (0,\infty)^4\) and \((\alpha,m)\in\mathbb{R}^2\). Here, \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^n\), \(n\ge 1\). When \(\chi>0\), it is shown that a non-negative global bounded classical solution exists provided that \(m>-2/n\), \(\beta>1\), and \(\alpha+\beta-m<(n+2)/n\). A similar result is obtained when \(\chi<0\), \(n\ge 3\), and \(g'\ge 0\) under the assumption \[ \alpha - m + \frac{n-2}{n+2} \beta < 1. \] In both cases, the proofs rely on energy-like estimates.
scientific article; zbMATH DE number 7595640
scientific article