Existence And Asymptotic Behavior Of Solutions To High-order Parabolic Equations

This thesis deals with existence and asymptotic behavior of solutions for multinonlinear high-order parabolic equations(systems).The substantial difficulty is that the general maximum principle does not hold any more for the high order cases.The topics include the large time behavior and the existence of solutions for a generalized thin film equation and a nonlinear fourth-order parabolic equation with gradient principal part,and the exponential decays for a viscous bipolar quantum hydrodynamic model with special third-order terms.Firstly,we consider a generalized thin film equation with zero-boundary fluxes.The decay of solution towards its mean is given by entropy functional method.Existence and positivity of solutions are studied for a thin film equation with a second-order diffusion term.Secondly,we concern the existence and asymptotic behavior of solutions for a nonlinear fourth-order parabolic equation with degenerate third-order derivative terms.Finally,we study the large-time behavior of solutions for a viscous bipolar quantum hydrodynamic model.The viscosity affects the speed of energy change,and specifically the bigger one yields the faster dissipation.Chapter 1 is to summarize the background of the related issues and to briefly introduce the main results of the present thesis.Chapter 2 is firstly concerned with the long time behavior of solutions to a class of fourth order nonlinear parabolic equation:u_t=-▽·(uⁿ▽△u +αu^n-1â–³uâ–½u+βu^n-2｜▽u｜²â–½u).The equation can be regarded as a generalization of the thin film equation u_t +(uⁿu_xxx)_x = 0 which is derived from a lubrication approximation and models the evolution of thin viscous films and spreading droplets.The function u represents thickness of the film.For the Neumann problem,we prove the algebra decay of solution towards its mean in L^∞-norm for the one-dimensional problem,and the exponential decay of solution to its average in L¹-norm for the multi-dimensional case,respectively.The main technical idea is to construct required dissipative entropies.We show that the derivative of entropy has a negative bound related to itself and for another the entropy has a L^p-norm low bound(0＜p≤∞).Thus,some long-time decay results can be obtained in the sense of L^p-norm.Secondly,we investigate nonnegative solutions of the thin film equation with a second-order diffusion term:u_t+(uⁿu_xxx)_x-(u^m)_xx= 0.For m＞n,existence of solutions is obtained in a weak sense.Positivity of solutions is collected,which is depending on n. Finally,we show that the classical solutions also converge to their mean at an exponential rate as the time t→∞.Chapter 3 is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear fourth-order parabolic equation:u_t+▽·(｜▽△u｜^P-2▽△u) = f(u) inΩ(?)R^N with boundary condition u =â–³u = 0 and initial data u₀.The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method.Unlike the general procedures used in the previous papers on the subject,we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables,respectively.By a compactness argument with necessary estimates,we show that the two approximation sequences converge to the same limit,i.e.,the solution to be determined.In addition, the decays of solutions towards the constant steady states are established via the entropy method.Finally,it is interesting to observe that the solutions just tend to the initial data u₀ as p→∞.Chapter 4 deals with large-time behavior of solutions for a viscous bipolar quantum hydrodynamic model.Generally,quantum effects,for example,resonant tunneling,are presented via microscopic models,such as the Wigner equation and the Schr(o｜¨)dinger system. These microscopic models can be described by macroscopic variables like current densities,electron densities and temperatures.QHD model,being as the extension of the classical hydrodynamic equations with quantum corrections,can be obtained from the Wigner-Boltzmann equation by employing the moment method or the Schr(o｜¨)dinger system.By applying the entropy method,we prove the exponential decays of solutions towards the constant steady states for the one-dimensional and the multi-dimensional cases.The argument is based on a series of a priori estimates.As a byproduct,the decay of solutions for the viscous hydrodynamic model is obtained as well.