Long-Time Behavior for a Thermoelastic Microbeam Problem with Time Delay and the Coleman-Gurtin Thermal Law
Wenjun Liu;Dongqin Chen;Zhijing Chen
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China
Microbeam problem;time delay;Coleman-Gurtin law;global attractor;exponential attractor
Acta Mathematica Scientia
National Natural Science Foundation of ChinaNational Natural Science Foundation of China (NSFC) [11771216, 11901306]; Key Research and Development Program of Jiangsu Province (Social Development) [BE2019725]; Natural Science Foundation of Jiangsu ProvinceNatural Science Foundation of Jiangsu Province [SBK2017043142]
This study addresses long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, the convolution kernel of which entails an extremely weak dissipation in the thermal law. By using the semigroup theory, we first establish the existence of global weak and strong solutions as well as their continuous dependence on the initial data in appropriate function spaces, under suitable assumptions on the weight of time delay term, the external force term and the nonlinear term. We then prove that the system is quasi-stable and has a gradient on bounded variant sets, and obtain the existence of a global attractor whose fractal dimension is finite. A result on the exponential attractor of the system is also proved.