Large Time Behavior Of Solutions To Hyperbolic Equations With Time-Dependent Damping

In this thesis,we consider large time behavior of solutions to two types of hyperbolic equations with time-dependent damping.Firstly,we consider large time behavior of solutions to the Cauchy problem for-system with time-gradually-degenerate damping term-_?1+?for 0?<1.Such a damping effect makes the original system possess the nonlinear diffu-sion phenomena time-asymptotic-weakly.By adopting the time-weighted energy method,where the weights are artfully chosen,we prove that the damped-system admits a unique couple of global solutions,and such solutions time-asymptotically converge to the shifted nonlinear diffusion waves.Furthermore,we get the con-vergence rates when the initial perturbations are in².The diffusion waves are the solutions of the corresponding nonlinear parabolic equation governed by the Darcy's law.Next,we consider large time behavior of solutions to the Cauchy problem with time-gradually-enhancing damping term-_?1+?for-1?<0.Firstly,we consider the case of-1<<0.We show the global existence and uniqueness for the smooth solutions.When the initial perturbations are in²,we derive the optimal convergence rates for the global smooth solutions to the nonlinear diffusion waves by selecting time-weighted functions technically.Subsequently,we consider the case of=-1.We also show that the damped-system has a unique couple of global smooth solutions,and the optimal convergence rate for the original solution to the shifted nonlinear diffusion wave is in the form of?1?ln^-34?2+?when the initial perturbations are in².Finally,we investigate large time behavior of solutions to the one-dimensional bipolar Euler-Poisson equations with time-dependent damping effect-_?1+?for-1?<1.Such a damping effect makes the original system possess the non-linear diffusion phenomena time-asymptotic-weakly or strongly.We divide the range ofinto four parts i.e.=-1,-1<<₇¹,=₇¹,₇¹<<1.By using the technical time-weighted energy method,where the weights are artfully chosen,we prove that the smooth solutions of the original system are unique and globally exist,and tend time-asymptotically to the corresponding diffusion waves,when the initial perturbations around the diffusion waves are small enough.When=-1,the convergence rate is?1?ln^-34?2+?.When??-1,₇¹?,the con-vergence rate is?1??1+?^-34^?1+?.When??₇¹,1?,the convergence rate is?1??1+?^-1.However,=₇¹is the critical point,and the convergence rate is?1??1+?^-67ln?2+?.All these convergence rates obtained as above are optimal when the initial perturbations are in².Particularly,when=₇¹,the conver-gence rate is the fastest,namely,the asymptotic profile of the original system at the critical point is the best.