Study On The Longtime Behaviors Of The Strain Solitary Waves In A Nonlinear Elastic Rod

In this doctoral dissertation, we have considered the long time behavior of the strain solitary waves for the nonlinear elastic rods with some suitable damped and the external forces f. That is, in mathematical setting, we have considered the existence of global attractors for the following equations whereΩ(?) R³ is an open bounded subset with smooth boundaryFirstly, in chapter 3, the existence and uniqueness of the global weak solution for the problem (κ) are proved by means of the Galerkin approximation method with the energy estimate, where the nonlinear term f satisfies a critical Sobolev exponential growth condition. These improved the related results, please see Theorems 3.2.3 and Theorems 3.3.1.Furthermore, in chapter 4, for the nonlinear term f which satisfies both conditions of subcritical and critical Sobolev exponential growth respectively, we have obtained the existence of global attractors of the semigroup{S(t)}_t≥0 for the wreak solution of (κ) in H₀¹(Ω)×H₀¹(Ω) respectively, by the use of theω-limit compactness methods and the asymptotical smoothness methods. This results can be fond in Theorem 4.1.3 and Theorem 4.2.6.Finally, in chapter 5, we proved the existence of global attractor of the semigroup {S(t)}_t≥0 for the strong solution in D(A)×D(A) by employing the method ofω-limit compactness, here the nonlinearity f with critical Sobolev exponential growth(see Theorem 5.1.3). It is valuable to notice that, we have established a analysis framework, namely Theorem 2.2.4 in chapter 2. Using the framework, we have proved the existence of a bounded absorbing set of the semigroup{S(t)}_t≥0 of the global strong solution for the wave equations(κ)in D(A)×D(A). Associated with this method, we efficiently conquered a difficulty in studying the global attractors of the semigroup for global strongly solution. The difficulty is that the nonlinearity with critical cannot be directly controlled by the linear term(see Theorem 5.1.2 and its proof). Additionally, we discussed the stability problem of system solution and obtained a sufficient condition which shows that the systems are eventually stable.