Long time behaviour of a Quasilinear wave equation

In this thesis, we concern the long time behaviours of the one dimensional strongly damped nonlinear wave equation u_tt-u_txx-∂/∂x (|u_x |^(p-2) u_x )+f(u)=g(x) (1) in bounded domain and the one dimensional strongly damped nonlinear wave equation with localized damping term u_tt-u_txx-u_xx-∂/∂x (|u_x |^(p-2) u_x )+a(x) u_t+f(u)=g(x) (2) in unbounded domain. The thesis consists of four sections. In the first section, the previous studies about the long time dynamics of the strongly damped wave equations and the main purpose of the thesis are mentioned. Second section is devoted to the main theorems and definitions. In the third section, we prove the well-posedness of the initial boundary value problem for equation (1) and the existence of regular strong global attractor in W_0^(1,p)(0,1)×L^2(0,1) for the semigroup generated by the problem. In the fourth section, we obtain the well-posedness of the initial value problem for equation (2) and the existence of weak local attractors in (W^(1,p)(R)∩H^1(R))×L^2(R) for the semigroup generated by the problem.