Abstract:

In this paper, we investigate the global existence of solutions to the following consumptive Keller-Segel model with singular sensitivity and porous medium diffusion $$\begin{align*} \left\{ \begin{aligned} &u_t=\Delta u^m-\chi\nabla\cdot(\frac{u}{v^\beta}\nabla v), \\ &v_t=\Delta v-vu^{\alpha}. \end{aligned}\right. \end{align*}$$ In the two dimensional space, it is shown that for any $m>1$, $\beta<\frac{11+8\sqrt 2}{28}(\approx 0.797)$, $\alpha<m+3(m-1)$, there exists a locally bounded global weak solution for any positive initial datum, furthermore, the solution is uniformly bounded in the sense of $L^p$-norm for any $p>1$. In the three dimensional space, it is shown that for any $m>\frac{10}9$, $\beta<\frac{3+\sqrt 3}{6}(\approx 0.789)$, $\alpha<\min\{\frac{32}5(m-1), m+3( m-\frac{10}9)\}$, there exists a locally bounded global weak solution, and the weak solution is uniformly bounded in the sense of $L^p$-norm for any $1<p<9(m-1)$. In addition, for any such solution, we prove that $v$ goes to zero uniformly as $t\to\infty$. It is worth noting that the global existence conclusion of the solution in this paper does not require any smallness restrictions on the initial values and parameters, thus expanding the scope of applicability of existing studies that rely on small initial values or small parameters.

Key words: singular sensitivity, local bounded global solution, long time behavior