Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities

Abstract

<p style='text-indent:20px;'>The chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{array}{l}\left\{ \begin{array}{l} u_t = \nabla \cdot \big( D(u) \nabla u \big) - \nabla \cdot \big( uS(x, u, v)\cdot \nabla v\big), \\ v_t = \Delta v -uv, \end{array} \right. \end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is considered in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n\ge 2 $\end{document}</tex-math></inline-formula>, with smooth boundary.</p><p style='text-indent:20px;'>It is shown that if <inline-formula><tex-math id="M3">\begin{document}$ D: [0, \infty) \to [0, \infty) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ S: \overline{\Omega}\times [0, \infty)\times (0, \infty)\to \mathbb{R}^{n\times n} $\end{document}</tex-math></inline-formula> are suitably smooth functions satisfying</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{array}{l}D(u) \ge k_D u^{m-1} \qquad {\rm{for\; all}}\; u\ge 0 \end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{array}{l}|S(x, u, v)| \le \frac{S_0(v)}{v^\alpha} \qquad {\rm{for\; all}}\; (x, u, v)\; \in \Omega\times (0, \infty)^2 \end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with some</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ \begin{array}{l}m>\frac{3n-2}{2n} \qquad {\rm{and}}\;\alpha\in [0, 1), \end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and with some <inline-formula><tex-math id="M5">\begin{document}$ k_D>0 $\end{document}</tex-math></inline-formula> and nondecreasing <inline-formula><tex-math id="M6">\begin{document}$ S_0: (0, \infty)\to (0, \infty) $\end{document}</tex-math></inline-formula>, then for all suitably regular initial data a corresponding no-flux type initial-boundary value problem admits a global bounded weak solution which actually is smooth and classical if <inline-formula><tex-math id="M7">\begin{document}$ D(0)>0 $\end{document}</tex-math></inline-formula>.</p>