This talk is concerned with the pantograph delay differential equation on $\mathbb{R}^d$ (d>1). First, the global wellposedness of the solution is proved. Then, a two-parameter semigroup on the weighted space is constructed, which is not formed naturally from the well-posedness of the solution due to the peculiarities of the delay term in the pantograph system. Finally, the existence of a pullback attractor and a forward attractor is established by the existence of a compact set which is uniformly attracting for the two-parameter semigroup associated to the system. The analysis of pantograph equations requires a non-autonomous set-up due to the special nature of the proportional delay, and remained as an open problem for more than 20 years. Thanks to a nice interpretation of the delay terms, we were able to handle the problem in an appropriate framework.
Tomás Caraballo教授，西班牙塞维利亚大学终身教授，博士生导师，美国奥本大学客座教授，主要研究随机偏微分方程吸引子的存在性和内部结构，随机动力系统等。先后在SIAM J. Math. Anal.，J. Differential Equations，Proc. A. J. Dynam. Differential Equations，Discrete Contin. Dynam. Systems等数学权威期刊上发表论文四百余篇，并著有《Applied Nonautonomous and Random Dynamical Systems, Applied Dynamical Systems》，多次受邀在华南理工大学，东华大学，华中科技大学等高校访问并做报告。
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