Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $ \mathbb{R}^{N} $
Authors: Shen Wenxian , Xue Shuwen 
Published: March 2022

<p style='text-indent:20px;'>The current paper is concerned with the spreading speeds of the following parabolic-parabolic chemotaxis model with logistic source on <inline-formula><tex-math id="M2">\begin{document}$ {{\mathbb R}}^{N} $\end{document}</tex-math></inline-formula>,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \begin{cases} u_{t} = \Delta u - \chi\nabla\cdot(u\nabla v)+ u(a-bu),\quad x\in{{\mathbb R}}^N, \\ {v_t} = \Delta v-\lambda v+\mu u,\quad x\in{{\mathbb R}}^N, \end{cases}\;\;\;\;\;\;\;\;\;\;\;\;\;\left(1\right) \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \chi, \ a,\ b,\ \lambda,\ \mu $\end{document}</tex-math></inline-formula> are positive constants. Assume <inline-formula><tex-math id="M4">\begin{document}$ b&gt;\frac{N\mu\chi}{4} $\end{document}</tex-math></inline-formula>. Among others, it is proved that <inline-formula><tex-math id="M5">\begin{document}$ 2\sqrt{a} $\end{document}</tex-math></inline-formula> is the spreading speed of the global classical solutions of (1) with nonempty compactly supported initial functions, that is,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \lim\limits_{t\to\infty}\sup\limits_{|x|\geq ct}u(x,t;u_0,v_0) = 0\quad \forall\,\, c&gt;2\sqrt{a} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \liminf\limits_{t\to\infty}\inf\limits_{|x|\leq ct}u(x,t;u_0,v_0)&gt;0 \quad \forall\,\, 0&lt;c&lt;2\sqrt{a}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M6">\begin{document}$ (u(x,t;u_0,v_0), v(x,t;u_0,v_0)) $\end{document}</tex-math></inline-formula> is the unique global classical solution of (1) with <inline-formula><tex-math id="M7">\begin{document}$ u(x,0;u_0,v_0) = u_0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ v(x,0;u_0,v_0) = v_0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ {\rm supp}(u_0) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ {\rm supp}(v_0) $\end{document}</tex-math></inline-formula> are nonempty and compact. It is well known that <inline-formula><tex-math id="M11">\begin{document}$ 2\sqrt{a} $\end{document}</tex-math></inline-formula> is the spreading speed of the following Fisher-KPP equation,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ u_t = \Delta u+u(a-bu),\quad \forall\,\ x\in{{\mathbb R}}^N. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Hence, if <inline-formula><tex-math id="M12">\begin{document}$ b&gt;\frac{N\mu\chi}{4} $\end{document}</tex-math></inline-formula>, the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation.</p>