Existence Of Solutions For A Class Of Higher Order Diffusion Equation

Nonlinear diffusion equation, as an important class of parabolic equations, comes from a variety of diffusion phenomena appeared widely in nature. They are suggested as mathematical models of physical problems in many fields such as filtration, phase transition, biochemistry and dynamics of biological groups. In the past several decades, specially in the recent 20 years, many researchers both in china and abroad have studied this kind of equations widely.In the first part, we consider the following initial and boundary value problem which comes from a population modelwhere B is the unit ball in R~2, n|→ is the outward unit normal to (?)B, k > 0, A(u) =u~3 -u.The above equation arises naturally as a continuum model for growth and dispersal in a population see[1], where u(x,t) denotesthe concentration of population, and the term g{u) is nonlinear function, denotes reaction term or power . Many authors have already studied the above problem, Liu and Pao [2] obtained the existence of classical solutions for the periodic boundary value problem;Chen Helv [3] obtained the existence and uniqueness of the classical solution for the initial and boundary value problem;Chen [4] obtained the existence for the Cauchy problem;and we have also proved the instability of the traveling wave solutions [5].In this paper, we study the radial symmetry solution for the above problem in two-dimensional space. After introducing the radial variable r = |rc|, we see that the radial symmetry solution satisfiesd(ru)dVdrdVr=o drr=l= 0,U=oGao and Yin [7] have studied the initial and boundary value problem of the radial solution for the following lubricant film equation(JiL— + div(m(u)VAu) = 0.Employing a similar method as [7], through considering the regularized problem, and by using some prior estimates, we establish theexistence of weak solution. It is easy to prove the existence of the classical solution for the regularized lubricant film equation which is considered by Gao and Yin. In order to solve our problem, we use a similar method as [6]. After establishing the regularization and the existence of the classical solution, we use some necessary consistent estimation to establish the existence of weak solutions.In the second part, we consider the Cahn-Hilliard equation with convectiondu dAu d2A{u) dB{u) _Where A{u) — 72U3 + 71W2 - u, B(u) = -j^4 + -w2, and 7 > 0>7i)72 is constant. According to the physical reality, we consider the equation (1) with the following boundary condition8ii du,t) = ?(l,t) = ^(0,t) - ^(l,t) = 0, t > 0. (2)As well known that the boundary condition is reasonable for the membrane equation or the Cahn-Hilliard equation ([23], [24], [25]). Moreover, the initial condition can be writen asu(x,O) = uo{x). (3)The equation(l) is derived from the continual model of the crystal growth process when.central plane and the angle raising([8], [9]), where u(x,T) denotes crystal plane slope. Convectiondu item (ii3 — u)— ([9]) comes from dynamics influence ( atoms ormolecules which have limited speed when stick to the crystal plane). It provides an item independent of the parameter, and is similar to the function of the exterior field when the pitch line of the driving system decomposes the pitch line.In past years, many scholars have paid attention to the following Cahn-Hilliard equation+ kAu AA{u), k>0, (4)C. M. Elliott & S. M. Zheng [10] have considered the initial and boundary value problem. Using the global energy estimates, for the classical case of A(u), the authors have proved the existence and uniqueness of the classical solution when N < 3 and the cubic term's coefficient is a positive constant or a negative constant with small initial energy. In addition, they also proved the blow-up property when the cubic term's coefficient is negative constant with large initial energy. Afterwards, some other properties of solutions have also been studied for Cahn-Hilliard equation with constant transport ratio, such as gradation [11], [12], [13], [14], [15], perturbation nature [ 16 ], [ 17 ], the global attractor's property [18], [19] , similar solution [20], and so on. However, only a few papers consider the Cahn-Hilliard equation of transfer type. It was K. H. Kwek [21] who first studied the equation for a special case with a special convection, namely, B(u) = u. Based on the discontinuous Galerkin finite element method, he proved the existence of classical solutions.Gao and Liu [22] have proved the instability of the traveling wave solution when B(u) = u. Using the energy estimates method, we shall prove the existence of the classical solution for the problem (1) - (3). In order to obtain some essential estimates, the Nirenberg inequality and Poincare inequality are used. Moreover, we also obtain the asymptotic property for the classical solution of problem(1) " (3).