Doğrusal olmayan disipatif denklemlerin çözümlerinin global davranışı

In this thesis, we investigate the global behavior of solutions of initial-boundary value problems (IBVP) for nonlinear dissipative equations. We particularly focus on two problems in hydrodynamics: IBVP for Burgers' original model of turbulence original Burgers' equations, and IBVP for Burgers' equation with nonlocal nonlinearity. Motivated by the studies in finite-dimensional asymptotic behavior of dissipative equations, we prove the stabilization of these equations by using finitely many controllers, such as finitely many Fourier modes, finitely many volume elements and finitely many nodal values. We also prove that the asymptotic behavior of solutions of original Burgers' equations can be completely determined by finite number of determining modes. Additionally, we show the existence, uniqueness and stability of the solutions of the inverse source problem for both equations. We show that, under proper assumptions, the solutions of the inverse source problem tends to a particular stationary state solution of the direct problem, and the unknown source term tends to zero as time goes to infinity. Finally, we perform numerical experiments to verify the validity of our theoretical findings on the finite-parameter feedback control problems for original Burgers' equations.