Asymptotic Limits On Nonlinear Partial Differential Equations

The introduction is located in Chapter one.In Chapter two,we study the semiconductor bipolar QHD model.First the existence of the thermal equilibrium solutions and the semiclassical limit results are obtained through the energy estimating method.In the evolutionary case the global existence and uniqueness for smooth solutions,the stability of the constant steady state solutions,the scmielassical limits and relaxation limits are established.The estimates on algebraic time-decay rate to the global solutions are proved by using the energy estimating method.In Chapter three we consider the generalized drift-diffusion model for semiconductors in the case of the electric field being vanished.The existence and uniqueness of the global smooth solution with owning the convergent self-similarity limits in the one dimensional case as well as the algebraic time-decay rate of the solution to the self-similarity solution are all given.In Chapter four we investigate the global smooth solutions and asymptotic limits of the isentropic compressible Navier-Stokes-Poisson equations.The L~2 estimates and the point-wise estimates are established through employing the L~p Fourier multiplier theory and the energy estimating method.The global existence and uniqueness are also obtained.Comparing with the general isentropic compressible Navier-Stokes equations we get the influence of the electric field on the decay rate of the global solutions of the Navier-Stokes equations which implies the both density time-decay rates same but the rate for momentum slower.In Chapter five we are concerned with the eigenvalue problem of an elliptic operator defined on the compound domain.We extend the results of Xuefeng Wang etc on the Dirichlet boundary conditions to the case of the third boundary conditions.We mainly investigate the influence of the three parameters,i.e.,the thermal conductivity coefficient in the coating body,the thickness of the coating body,and the thermal exchange coefficient in the boundary conditions,on the eigenvalue estimates. In Chapter six the asymptotic limits of the elliptic third boundary value problem are analyzed.We prove the asymptotic limits results similar to the ones in the first boundary value by Caffarelli and Friedman hold in the case of the third boundary value.Some new estimates on the higher order derivatives and some other conclusions are obtained.