Existence And Asymptotic Behavior Of Solutions For Several Kinds Of Thermoelastic Microbeam System

In the present years,the research of coupled thermoelastic vibration characteristics for engineering system components has become a new research field.Euler-Bernoulli beam’s model is the most commonly used model.Based on this model,microbeam as critical components in microelectromechanical systems has considerably aroused the interest of researchers.This paper mainly considers the well-posedness and long time behavior of microbeam system with different thermal laws.The rest of our paper is organized as follows:In Chapter 1,we review the background and development of the related systems,and the brief intruduction of this paper.Some denotes and results are presented.In Chapter 2,we study the well-posedness and the asymptotic stability of a one-dimensional thermoelastic microbeam system,where the heat conduction is given by Gurtin-Pipkin thermal law.We first establish the well-posedness of the system by using the semigroup arguments and Lumer-Phillips theorem.We then prove a general decay of the total energy of our system,which can recover the exponential rate in the special case.In Chapter 3,we study the long-time behavior for a thermoelastic microbeam problem that incorporates time delay and Coleman-Gurtin thermal law.We consider here a convolution kernel 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,under suitable assumptions on the weight of time delay term,the damping term and the external force term.Since the system is quasi-stable on bounded variant sets,we then prove that it is asymptotically smooth.By using the gradient system and asymptotic smoothness of the system,we obtain the existence of a compact global attractor,which has finite fractal dimension.Result on the exponential attractor of system is also proved.In Chapter 4,we study the long-time behavior of a semilinear microbeam problem with only one nonlinear term and with the dissipation due to the thermal and mass diffiusion.The thermal and mass diffusion conduction are modeled by the Gurtin-Pipkin law in presence of time-dependent memory kernel.We first prove the well-posedness by the semigroup theory.And then,by using the gradient system and asymptotic smoothness of the system,we prove the existence of a global attractor,which are characterized as unstable manifold of the set of stationary solutions.We use multiplier method to establish a stabilizability inequality to get the quasi-stability of the system and prove that the global attractor has finite fractal dimension.In Chapter 5,we summarize the context of this paper and introduce the prospect of future research.