General Stability Of Abstract Thermoelastic System With Infinite Memory And Delay

As an important branch in the field of modern mathematics,the theory of partial differential equations is widely used in many fields of science and engineering,such as physical chemistry,ecological and economic systems and atmospheric space science.At the same time,with the development of the research,the problems previously approximated by a single differential equation now must also be taken into account the influence of more other factors.Therefore,the research on coupled partial differential equation has shown great theoretical and practical value which has attracted many scholars to carry out a series of in-depth research on this.In recent years,the research on thermoelastic system has attracted the attention of many scholars,which considers the results of the interaction between temperature and deformation,that is,not only temperature will produce deformation,but also deformation will produce or consume energy and affects the temperature.From a large number of research results,it can be known that time delay is the source of instability,so how to add new conditions to reduce the influences of time delay and restabilize a system become an important research direction in the field of partial differential equations.In this thesis,we mainly study the well-posedness and stability of abstract thermoelastic systems with infinite memory and delay.In the first chapter,we present the physical background of thermoelastic system,and introduce the research status of some evolution equations and equations with time delay and the main problem will be investigate.In the second chapter,we give some necessary assumptions and by constructing an appropriate state space to get a new system which equivalent to the original system,then we also prove the well-posedness of system by invoking semigroup theory.In the third chapter,we investigate the general stability of system by using multiplier method and constructing suitable Lyapunov functionals which are equivalent to the energy functional in two cases p=1 and 1<p<3/2 respectively.In the fourth chapter,we give some applications of stability for thermoelastic systems with infinite memory and delay.