报告题目：Longtime dynamics of a class of quasi-linear wave equations

报告摘要：In this talk, we investigate the well-posedness and the longtime dynamics of a class of quasilinear wave equations with structural damping $u_{tt}+(-\Delta)^\alpha u_t-\Delta u +f(u)=\nabla\cdot\phi^\prime (\nabla u)+g$, with $1/2<\alpha\leq1$.  (i) When $\alpha=1$, we prove that the energy solution of the equation is stable and the related dynamical system possesses a (strong) global attractor in natural  energy space  (rather than a weak one as known before). (ii) When $1/2<\alpha<1, \phi^\prime (\eta)=\frac{\eta}{\sqrt{1+|\eta|^2}}$, the equation becomes  $u_{tt}-\Delta u+(- \Delta)^{\alpha}u_t-\nabla\cdot  \Big(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \Big)+g(u)=f(x)$.  The main results are concerned with the quasilinear term $\nabla\cdot  \Big(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \Big)$ and the nonlinearities $g(u)$ with supercritical growth. Under the rather mild conditions, the well-posedness and the existence of the global and exponential attractors (rather than the weak ones) are established in natural energy space.  These results show that even for the supercritical nonlinearities, the regularity and the longtime dynamics of the solutions of the mentioned equations are of the characteristics of the parabolic equations because of the effectiveness of the structural damping. 

报告人简介：杨志坚，郑州大学理学博士，日本九州大学数理学博士，郑州大学2级教授、博士生导师, 河南省跨世纪学术技术带头人，河南省数学会常务理事。现任美国《Mathematical Reviews》评论员,《Journal of Partial Differential Equations》编委。杨志坚教授主要研究出自物理、力学和量子力学中的非线性发展方程的适定性及对应的无穷维动力系统的长时间行为。主持国家自然科学基金面上项目3项；在国外SCI期刊《J. .Differential Equations》、《Nonlinearity》、《Discrete Contin. Dyn. Syst.：A》、《Commun. Contemp. Math.》、《Appl. Math. Lett.》、《JMAA》、《Nonlinear Anal.》、《Dynamics of PDE》、《J. Math. Phys.》等发表论文60多篇。主持完成河南省自然科学基金项目6项。获得2000年河南省科技进步二等奖一项。

报告时间：2017年12月8日（星期五）上午10: 00-11：00               

报告地点：科技楼602报告厅