Study Of An Initial-Boundary Value Problem For Nonlinear Thermoelastic Beam Equations With Second Sound

In this master thesis, we are going to study the well-posedness and the asymptotic behavior of the solution to thermoelastic beam equations with second sound. In the one-dimensional case, this system describes the interrelation among vertical deformation, temperature difference and heat flux. The deformation and the temperature difference are coupled. And the relationship between temperature and heat flux is governed by Cattaneo’s Law which will lead to the propagation speed of temperature being finite.In the section of introduction, we will review the research history and current status for beam equations and the coupled systems with heat equation. These systems can be either classical thermoelastic equations or thermoelastic system with second sound.In the second chapter, we will first study the well-posedness and energy estimates of a linearized problem for the thermoelastic beam equations with second sound. Based on the dual argument and the Fredholm Alternative Theorem, we obtain the existence of the weak solutions of the linearized problem. Then with the method of energy estimates, under the compatibility conditions, the classical solution of the linearized problem is obtained.In the third chapter, we will mainly study the existence of global classical solutions for the nonlinear thermoelastic system with second sound. After linearizing the nonlinear problems and using the conclusion of the second chapter, we establish the well-posedness and energy estimates of a sequence of approximate solutions and then get the existence of the classical solution for nonlinear problems by the iterative method and the priori estimates.In the fourth chapter, we discuss the energy behavior of the nonlinear thermoelastic system with second sound when Ï„, the relaxation factor in Cattaneo’s law, tends to zero. On the basis of certain compatibility conditions for initial data, through a series of estimation, we deduce that the energy converges to the corresponding energy for classical thermo elasticity system at the rate of Ï„2 as Ï„â†' 0.Finally, in the appendix, we are using the operator decomposition method and the semigroup method to discuss the well-posedness theory of the initial boundary problem for beam equations.