Stability Analysis Of One-Dimensional Linear Mixed Dynamic Thermoelastic System

Based on the three non-classical thermoelastic theories proposed by Green and Naghdi,we discuss the energy decay rate of the 1-D linear mixed thermoelastic system.The main contents are summarized as follows:We prove that the energy of the mixed type I and type II thermoelastic system can not achieve exponential stability,but under the smooth initial condition,the system is polynomially stable with optimal decay rate_t~1,that is,it can not be improved.Fur-thermore,when the viscous damping enters in the mixed type II and type III system,based on the position of type III thermoelastic area relative to the whole area,two class-es of energy decay rates are obtained.When type III thermoelastic region includes one end point of the whole region,the system can achieve exponential stability.Otherwise,when the type III thermoelastic region is strictly located in the whole domain,the sys-tem lacks of exponential decay rate,but the polynomial decay holds and the sharpness of optimal polynomial decay order is further verified..Stability of the system is mainly studied by the frequency domain method.Firstly,we establish the appropriate state space and define the corresponding system operator that rewrites the system into the form of Cauchy evolution equation.Then,the well-posedness of the dynamic system is discussed by semigroup theory.Secondly,based on the dissipativeness of the system operator,by judging whether there are spectrums of system operator on the imaginary axis,we prove the strong stability of the system.Furthermore,if the norm of the resolvent operator along the imaginary axis is bounded uniformly,the exponential decay rate of the energy in the system is showed in frequency domain method.If the resolvent estimate is unbounded,that is,the system can not reach exponential stability,the polynomial stability of the system is proved by a relatively weak resolvent estimate and the optimality of the polynomial decay rate is verified by constructing a counterexample.Finally,we present some numerical simulations on the dynamic behavior of the system to support the theoretical results by Matlab software.