Global classical solutions of a degenerate migration system with indirect signal absorption in arbitrary dimensions

Abstract

This paper deals with a no-flux initial–boundary value problem ($\star$)$\left(\begin{array}{c}{u}_t=\Delta \left(u\varphi \left(v\right)\right),\\ v_t=\Delta v-wv,\\ \tau w_t=\Delta w-w+u\end{array}\right)$in a bounded domain $\Omega \subset R^n(n\ge 2)$ with smooth boundary, where $\tau \in \{0,1\}$ and the nonnegative motility function $\varphi$ admits the degenerate case: $\varphi \left(0\right)=0$. It is shown that for any appropriately regular initial data, global classical solutions of ($\star$) can be constructed, if either $\tau =0$ or ${sup}_{\xi \in [0,{\Vert v_0\Vert }_{L^{\infty }\left(\Omega \right)}]}\varphi \left(\xi \right)<1$ when $\tau =1$.