The trichotomy of solutions and the description of threshold solutions for periodic parabolic equations in cylinders
报告人：王智诚（兰州大学 ）
时间：2022.5.13上午9：00-10：00
地点：腾讯会议，415 246 745
摘要：In this talk we consider the nonnegative bounded solutions for reaction-advection-diffusion equations of the form $u_{t}-\Delta u+\alpha(t,y)u_{x}=f(t,y,u)$ in cylinders, where $f$ is a bistable or multistable nonlinearity which is $T$-periodic in $t$. We prove that under certain conditions, there are at most three types of solutions for any one-parameter family of initial data: that spread to $1$ for large parameters, vanish to $0$ for small parameters, and exhibit exceptional behaviors for intermediate parameters. We usually refer to the last as the threshold solutions. It is worth noting that we also give a sufficient condition for solutions to spread to $1$ by proving a kind of stability of a pair of diverging traveling fronts. Furthermore, under the additional conditions, by using super- and sub-solutions, Harnack's inequality and the method of moving hyperplane, we show that any point in the $\omega$-limit set of the threshold solutions is symmetric with respect to $x$, and exponentially decays to $0$ as $|x|\to\infty$.
专家介绍：王智诚，兰州大学数学与统计学院教授，甘肃省飞天学者特聘教授，博士生导师。1994年本科毕业于西北师范大学，2007年在兰州大学获理学博士学位，2008年3月至2009年3月在加拿大约克大学从事博士后工作一年，2014年到法国波尔多大学访问。在Trans. AMS、SIAM J. Math. Anal.、SIAM J. Appl. Math.、JMPA、Calc. Var. PDE、JDE、JDDE、Nonlinearity、J. Math. Biol.、J. Nonlinear Sci、Proc. Royal. Soc. A等杂志发表SCI论文80多篇。2010年入选教育部新世纪优秀人才支持计划，2011和2019年分别获得甘肃省自然科学二等奖，2016年入选甘肃省飞天学者特聘教授，主持完成两项国家自然科学基金面上项目以及教育部博士点基金等多项省部级项目，正在参加一项国家自然科学基金重点项目。目前担任两个SCI杂志International J. Bifurc. Chaos 和Mathematical Biosciences and Engineering (MBE) 的编委（Associate editor）。
