Global generalized solvability in the Keller-Segel system with singular sensitivity and arbitrary superlinear degradation

Abstract

<p style='text-indent:20px;'>This paper considers the Neumann initial-boundary value problem for the chemotaxis system with singular sensitivity</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{split} \left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \chi\nabla \cdot (\frac{u}{v}\nabla v) + f(u),}&amp;{x \in \Omega ,t &gt; 0,} \\ {{v_t} = \Delta v - v + u,}&amp;{x \in \Omega ,t &gt; 0,} \end{array}} \right. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a smooth bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega \subset {\mathbb{R}^{n}} $\end{document}</tex-math></inline-formula><inline-formula><tex-math id="M2">\begin{document}$ (n\geq2) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ f\in C^{1}([0,\infty)) $\end{document}</tex-math></inline-formula> generalizes the logistic function <inline-formula><tex-math id="M4">\begin{document}$ f(s) = \lambda s-\mu s^{\alpha} $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M5">\begin{document}$ \lambda\geq 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \mu&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \alpha&gt;1 $\end{document}</tex-math></inline-formula>. We prove global existence of solutions to this system in an appropriately generalized sense for any <inline-formula><tex-math id="M8">\begin{document}$ \chi&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \alpha&gt;1 $\end{document}</tex-math></inline-formula>.</p>