Duan Wu

The present work proceeds to consider the convergence of the solutions to the following doubly degenerate chemotaxis-consumption system $ \begin{align*} \left\{ \begin{array}{l} &u_{t} = \nabla\cdot\big(u^{m-1}v{\nabla u}\big)-\nabla\cdot\big(f(u)v\nabla v\big)+\ell uv, \\ &v_{t} = \Delta v-uv, \end{array}\right. \end{align*} $ under no-flux boundary conditions in a smoothly bounded convex domain $ \Omega\subset \mathbb{R}^2 $, where the nonnegative function $ f\in C^1([0, \infty)) $ is asked to satisfy $ f(s)\le C_fs^{\alpha} $ with $ \alpha, C_f>0 $ for all $ s\ge 1 $. The global existence of weak solutions or classical solutions to the above system has been established in both one- and two-dimensional bounded convex domains in previous works. However, the results concerning the large time behavior are still constrained to one dimension due to the lack of a Harnack-type inequality in the two-dimensional case. In this note, we complement this result by using the Moser iteration technique and building a new Harnack-type inequality.