This work considers the doubly degenerate nutrient model \begin{equation*}\label{AH1} \left\{ \begin{split} &u_t=\nabla\cdot\left(u^{m-1}v\nabla u\right)-\nabla\cdot\left(f(u)v\nabla v\right)+\ell uv,&&x\inΩ,\,t>0, &v_t=Δv-uv, &&x\inΩ,\,t>0, \end{split} \right. \end{equation*} under no-flux boundary conditions in a smoothly bounded convex domain $Ω\subset \mathbb{R}^n$ ($n\le 2$), where the nonnegative function $f\in C^1([0,\infty))$ is assumed to satisfy $f(s)\le C_fs^α$ with $α>0$ and $C_f>0$ for all $s\ge 1$. When $m=2$, it was shown that a global weak solution exists, either in one-dimensional setting with $α=2$, or in two-dimensional version with $α\in(1,\frac{3}{2})$. The main results in this paper assert the global existence of weak solutions for $1\le m<3$ and classical solutions for $3\le m<4$ to the above system under the assumption \begin{equation*} α\in \left\{ \begin{split} &\left[m-1,\min\left\{m,\frac{m}{2}+1\right\}\right]~~&&\textrm{if}~~n=1,\quad\quad\textrm{and} &\left(m-1,\min\left\{m,\frac{m}{2}+1\right\}\right)~~&&\textrm{if}~~n=2, \end{split} \right. \end{equation*} which extend the range $α\in(1,\frac{3}{2})$ to $α\in(1,2)$ in two dimensions for the case $m=2$. Our proof will be based on a new observation on the coupled energy-type functional and on an inequality with general form.
On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport
本文提供了一个统一的数学分析家庭的非局部扩散界面模型的肿瘤生长描述的演变驱动的长程相互作用。这些积分-偏微分方程通过一个非局部项来模拟细胞间粘附，可以看作是H. Garcke et al.（2016）.该模型考虑耦合的非本地Cahn-Hilliard方程的肿瘤相变量与营养浓度的反应扩散方程，并考虑到重要的机制，如趋化性和主动运输。该系统依赖于两个松弛因子：化学势上的粘性系数和抛物线正则化系数。本文的第一部分是专门分析的系统与两个正则化。在这里，一个丰富的频谱的结果。弱适定性首先解决，也包括奇异的潜力。在适当的条件下，证明了具有分离性的强解的存在性.这也允许获得关于数据的精确稳定性估计，包括趋化性和主动转运。第二部分研究了当松弛因子为零时系统的渐近行为。当参数分别和联合趋近于零时，分析了渐近性，并得到了精确的误差估计。作为副产品，相应的极限系统的适定性建立。
