報告題目：Some progress for the global existence and boundedness of high-dimensional chemotaxis-haptotaxis models with re-establishment mechanisms
報 告 人：鄭甲山 教授 煙台大學
報告時間：2024年5月21日 9:00-10:00
報告地點：騰訊會議 ID：895-979-166
或點擊鍊接直接加入會議：https://meeting.tencent.com/dm/whpW81vF2ke1
校内聯系人：劉長春 liucc@jlu.edu.cn
報告摘要：
The chemotaxis--haptotaxis model with remodeling of non-diffusible attractant
$$
\left\{\begin{array}{ll}
u_t=\Delta u-\chi\nabla\cdot(u\nabla v)-
\xi\nabla\cdot(u\nabla w)+f(u,w),\\
\disp{v_t=\Delta v- v +u},\quad
\\
\disp{w_t=- vw+\eta w(1-u-w),}\quad\\
\end{array}\right.
$$
is considered in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary, where $\chi >0, \xi >0$ as well as $\eta > 0$ are given parameters. This model is initially proposed by Chaplain and Lolas (2006) \cite{Chaplain7} to describe the interactions between cancer cells, the matrix degrading enzyme and the host tissue in a process of cancer cell invasion of tissue
(extracellular matrix). Assume that $f(u,w)=\mu u(1-u-w)$.
The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded classical solutions to an associated initial-boundary value problem under the assumption that $\mu$ is appropriately large. After conducting thorough research and incorporating the essence of numerous prior studies, this paper not only extends the findings of these previous works (see {Remark} 1.1) but also deepens our understanding of chemotaxis-haptotaxis models. Notably, we have successfully demonstrated for the first time the boundedness of solutions in a three-dimensional chemotaxis-haptotaxis model featuring the remodeling of non-diffusible attractants. This significant discovery undoubtedly adds new dimensions to the theoretical framework of chemotaxis-haptotaxis models. Furthermore, the achievement of this milestone not only broadens the scope of research in chemotaxis-haptotaxis models but also provides researchers in related fields with fresh perspectives and ideas, paving a new path for future studies. At the same time, some extensions will be made to this model, and some methods of this model will be used to summarize and promote relevant models.
報告人簡介：
鄭甲山，煙台大學數學與信息科學學院教授，碩士生導師, 山東省傑出青年基金和山東省優秀青年基金獲得者，獲得首屆山東數學會青年數學獎。主要面向生物科學與力學及物理學、醫學與流體動力學等領域偏微分方程的數學問題，主要開展趨化-（納維）-斯托克斯相關模型、非線性抛物型方程與流體動力學方程等學科領域的熱點問題研究。主持（完成）山東省傑出青年基金、山東省優秀青年基金、國家自然科學基金、中國博士後特别資助和博士後面上資助、山東省自然科學青年基金等多項基金。并以第一或者通訊作者在CVPDE、M3AS、JDE、Nonlinearity等高水平期刊上發表SCI論文70餘篇，包含4篇 ESI 高被引論文，連續三年入選斯坦福大學發布的“全球前2%頂尖科學家榜單”。已被包括國際數學家大會45分鐘報告人、長江學者特聘教授、頂級期刊《M3AS》主編、《JDE》等著名雜志編委在内的多名數學專家引用總次數800餘次。應國際物理科學院院士Hari M. Srivastava教授所邀在 Springer雜志合作撰寫趨化-N-S相關模型的專著。應邀擔任國際期刊《American Journal of Applied Mathematics》、《Mathematics and Computer Science 》和《World Journal of Mathematics and Statistics》和《Applied and Computational Mathematics》的編委；應邀參加中國數學會第十三次全國代表大會并作報告；應邀擔任美國《Mathematical Reviews》評論員和德國《數學文摘》評論員。
