Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion

Abstract

<p style='text-indent:20px;'>We consider a parabolic-elliptic system of partial differential equations with a chemotactic term in a <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-dimensional unit ball "<inline-formula><tex-math id="M2">\begin{document}$ B $\end{document}</tex-math></inline-formula>" describing the behavior of a biological species "<inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>" and a chemical stimuli "<inline-formula><tex-math id="M4">\begin{document}$ v $\end{document}</tex-math></inline-formula>". The system presents a sub-linear dependence of "<inline-formula><tex-math id="M5">\begin{document}$ \nabla v $\end{document}</tex-math></inline-formula>" in the chemotactic coefficient and a nonlinear diffusive term. The evolution of <inline-formula><tex-math id="M6">\begin{document}$ u $\end{document}</tex-math></inline-formula> is described by the equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t - \Delta u^m = - div (\chi u |\nabla v|^{p-2} \nabla v), \quad \mbox{ for } \ m &gt;2, \quad p \in ( 1,2), \quad N \geq 1 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for a positive constant <inline-formula><tex-math id="M7">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>. The concentration of the chemical substance <inline-formula><tex-math id="M8">\begin{document}$ v $\end{document}</tex-math></inline-formula> satisfies the linear elliptic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ - \Delta v = u - \frac{1}{|B|} \int_{B} u_0dx. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We consider the radially symmetric case and we prove the local existence of weak solutions for the mass accumulation function under assumption</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ - \frac{1}{m}+ \frac{1}{N} + 1-\frac{pm}{4(m-1)} \geq 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for radial and regular initial data. Additionally, if the constrain</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ \frac{m }{m- 2} \left[ \frac{pm}{2(m-1)}-1\right] \leq 1 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is satisfied, the solution globally exists in time.</p>