Some Problems For A Class Of Fourth Order Parabolic Equations

The fourth-order parabolic partial differential equations have many important applications in the fields of image processing,materials science,engineering,biological mathematics,etc.Recently,many authors have theoretically investigated the fourth-order parabolic partial differential equations.In this thesis,we consider a fourth-order low-curvature equation,film equation with logarithmic nonlinearity and a class of degenerate fourth-order parabolic equation.Firstly,we discuss a fourth-order parabolic initial-boundary value problem,which is of the form:where ?(?)R2 is a bounded open domain with smooth boundary(?)?,T is a positive number and p>2.By employing the difference method,the evolution equation can be written as elliptic equations,which possess the existence and uniqueness of the weak solutions.By approximation we can obtain the weak solutions of the original evolution problem.Secondly,we discuss a nonlinear parabolic initial-boundary value problem as fol-lows:where ?(?)Rn(n? 1)is a bounded domain,its boundary(?)? is smooth,and v is the outward normal on(?)?,u0?H02(?)By combining the classical Galerkin's method with the modified potential well method,we investigate the existence and uniqueness of global weak solution,the blow-up in finite time of solutions,and the decay rate for global solutions according to three different initial data.Finally,we investigate a class of fourth-order parabolic equation modeling Bose-Einstein particles:This model is a Fokker-Planck approximation of Boltzmann-Nordheim function,and maintains some important features of the original Boltzmann equation.Due to the degeneration,we first establish the existence of classical solution for non-degenerate approximate problem,and then obtain the existence of local solution for original prob-lem by utilizing the uniform estimation of the approximate solutions.