| The phase transition is an important class of diffusion phenomena. In the phase transition theory, one proposed many important models of higher order diffusion such as Kuramoto-Sivashinsky equation,thin film equation, Cahn-Hilliard equation, Cahn-Hilliard-Gurtin equation (see [l]-[3]). In the past decades, this class equation have been widely by a large number of mathematicians both in China and abroad.In this paper, we focused to study a degenerate higher order equation. It was F.Bernis and A.Friedman [4] who first study the degenerate higher order equation. They consideblack the thin film equation with initial-boundary value problemwhere n≥ 1. The authors proved the existence of generalizedsolution. Deal Passo, Garcke and Grim [5] have obtain the similar results by using energy and global entropy estimates for the two dimensional case if n > -;and the three dimensional case iflAs another important kind of fourth order equation, the Cahn-Hillaird equation was originally proposed by Cahn and Hilliard in 1958 as a model of spinodal decomposition for a binary mixtures. It was also derived from competition and exclusion of biological groups, moving process of river basin, diffusion of oil film over a solid surface. Yin [6] consideblack following Cahn-Hilliard equation with initial-boundary value conditionsill I— + D[m(u)(KD3u - DA(u))] = 0, Du1=0,11=0,1and proved the existence and nonnegative property of weak solution. Later on, J.R.King[5] proposed a generalized thin film equationt + div(un\VAu\p-2VAu) = 0, xett,t>0,p>2. otThe equation describes the surface tension driven evolution of the height u(x, t) of a thin liquid film on a solid surface in lubrication approximation]^, 6, 7]. The exponent p is related to the rheologicalproperties of the liquid: p = 2 corresponds to a Newtonian liquid, whereas p ^ 2 emerges when considering "power-law" liquids, when p > 2 the liquid is said to be "shear-Thinning".Liu, Yin and Gao [9] discussed the following equation with initial-boundary value conditionsill I— + divfl VAu|p~2VAu) = 0, xeti,t>0,p>2,u = Aw = 0, x G <9ft, t>0, u(x, 0) = uo(x), x E Q.The existence of weak solutions was established by the time-discrete method for the two dimensional case. The uniqueness and asymptotic behavior of solutions are discussed. Liu[10] discussed the finite speed of propagation of perturbation and regularity of solutions for the one-dimensional case.Xu and Zhou[ll] consideblack the following initial-boudary value problem— + div(\VAu\p~2VAu) = f - divg, x G ft,u = o, Au = o, x € on,u(x,0) = wo(ar), x € ft,where the cylinder Q = ft x (0, T], the lateral surface T = <9ft x (0, T\. The exitence and uniqueness of weak solutions were investigated by employing the difference and variation method. Some regularity results of weak solutions were deduced.In this paper, we consider following initial-boundary value problem— + A(\Au\p-2Au) + X\u\p~2u = 0, x € ￡l,p > 2, A > 0, (1) ot= Au\dn = 0, (2)udnu(x,0) = uq(x), iefi. (3)where fi is a bounded domain in M.N, A2 = A(|Au|p~2Au) is said to be a p-biharmonic operator, it is called ordinary biharmonic operator when p = 2. when p = 2, A = 0, the case is the sane as equation (1) when p = 2. It is noted by many scholars both in China and abroad for the p-biharmonic equation. Jiff Benedikt [12] studied the p-biharmonic equation A(|Ait|p~2Au) = \\u\q~2u, A € R,p,q > 1, the existence is proved for p > q for the initial value problem,the un-queness is proved for p < q. Pavel Drabek and Mitsuharu Otani [13] proved that the nonlinear eigenvalue problem for the p-biharmonic operator with p > l,and Q, is a bounded domain in RN with smooth boundary,has principal positive eigenvalue which is simple and isolated.In this paper, we prove the exisstence of-the problem (1)-(3). First, we use the time-descrete method for constructing an approximate solution, hence, e establish a priori estimates of the approximate solution, we prove the existence subsequently by taking the limit of the approximate solution,then we have following results namelyTheorem 1. Let uo ￡ Wq'p(Q), p > 2, then the problem (l)-(3) admit at least one weak solution.We also prove the uniqueness of the weak solution and discuss the properties of the weak solution. Using Poincare inequality, Nirenberg inequality and Hardy inequality, we prove the uniqueness of the weak solution, discuss the finite speed of propagation of perturbation and asymptotic behavior of the weak solution. Our main results areTheorem 2. Assume p > 2 and u is the weak solution of the problem (l)-(3), then we have/ C2 Ci ^Jn (Oir + G2)a p-2Theorem 3. Assume p > 2, |crn(0)| < b and u is the weak solution of problem (l)-(3), then for any fixed t > 0, we have0rn{t) 0, /3 > 0,6 > 0 are constants independent of t. |
