The asymptotic behavior of solutions to a doubly degenerate chemotaxis-consumption system in the two-dimensional setting
Duan Wu
TL;DR
We study the long-time behavior of the 2D doubly degenerate chemotaxis–consumption system $u_t=\nabla\cdot( u^{m-1}v\nabla v)-\nabla\cdot( f(u) v\nabla v)+\ell uv$, $v_t=\Delta v-uv$ in a smooth convex domain $Ω⊂ℝ^2$. The authors combine a regularization, Moser iteration to obtain uniform $L^{\infty}$ bounds for $u$, and a new Harnack-type inequality for $v$, followed by a time-rescaling that yields a porous-medium–type limit with uniformly elliptic coefficients. Using compactness (Arzelà–Ascoli) and Schauder/Hölder theory, they obtain $u(\cdot,t)\to u_∞$ in $L^{\infty}(Ω)$ and $v(\cdot,t)\to0$ as $t\to∞$, with $u_∞$ characterized as $w(\cdot,1)$ where $w$ solves the transformed weak problem. The results provide a rigorous 2D asymptotic description for this class of cross-diffusion systems and introduce a methodological route potentially applicable to related models.
Abstract
The present work proceeds to consider the convergence of the solutions to the following doubly degenerate chemotaxis-consumption system \begin{align*} \left\{ \begin{array}{r@{\,}l@{\quad}l@{\,}c} &u_{t}=\nabla\cdot\big(u^{m-1}v\nabla v\big)-\nabla\cdot\big(f(u)v\nabla v\big)+\ell uv,\\ &v_{t}=Δv-uv, \end{array}\right.%} \end{align*} under no-flux boundary conditions in a smoothly bounded convex domain $Ω\subset \R^2$, where the nonnegative function $f\in C^1([0,\infty))$ is asked to satisfy $f(s)\le C_fs^{\al}$ with $\al, C_f>0$ 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.