Radially Symmetric Steady States Of The Cahn-Hilliard Equation

Cahn-Hilliard equations, as an important class of higher order nonlinear diffusion equations, come from a variety of diffusion phenomena in the nature. In particular, such equations are suggested as mathematical models of physical problems in many fields such as diffusion phenomena in phase transition [3], [5], [6], competition and exclusion of biological groups [7], moving process of river basin [8], diffusion of oil film over a solid surface [9]. In the last decades, especially in recent twenty years, the study in this direction attracts a large number of mathematicians both in China and abroad.In 1958, J. W. Cahn and J. Hilliard studied the diffusion phenomena in two different phases and came up with this type of equation. We just explain the physics background of Cahn-Hilliard type equation by this diffusion phenomena. The process of phase segregation following a quench (sudden cooling) of a system from high temperature, where the system has a unique uniform equilibrium phase, into the miscibility gap where two (or more) phases cancoexist is variously known as spinodal decomposition, nucleation, coarsening, etc.. It concerns the tendency of the different phases to segregate, creating larger and larger domains of approximately homogeneous single-phase regions. The problem is of great practical importance in the manufacturing of alloys, where the degree of segregation influences the properties of the material. The mathematical description of the time evolution of the local macroscopic order parameter in such systems, e.g., the difference in the concentration of A and B atoms in a binary A-B alloy, is commonly given by nonlinear fourth-order equations of the Cahn-Hilliard type.We will also show how to get the Cahn-Hilliard type equation in some simple cases by the process of phase segregation. The Ginzburg-Landau free energy of the mixture F is a functional of uwhere (u), H(u) are given. The corresponding potential function isWe usually suppose that the flux J is in proportion to the gradient of potential with reverse direction, namelyBy the mass balance, we haveThenParticularly when (u) is a constant, for example (u) = 1, The above equation can be written asThis is the Cahn-Hilliard equation with concentration dependent, where u denotes the concentration of one of two phases in a system, m(u) is the mobility, A(u) = -H'(u) is the intrinsic chemical potential and H(u) is the double-well potential. A reasonable choice of A(s) is the cubic polynomial, namelyIf the mobility is a constant in equation (1), without loss of the generality we assume m(u) = 1, then equation (1) is the Cahn-Hilliard equation with constant mobilityBy a radially symmetric solution u, we mean u(r,t) = u(|x|,t), r = |x|, which satisfieswhere . The associated free energy if where C is a constant depending only on the spacial dimension n. It is well known that the steady states of the equation minimizes the free energy F. In other words, the radially symmetric steady states satisfyThe solution of the equation (3) can also be regarded as the radially symmetric steady states of the Allen-Cahn equationIt was C.M.Elliott and S.M.Zheng [6] who first study theoretical the Cahn-Hilliard equations. They considered the Cahn-Hilliard equation with constant mobility. Basing on global energy estimates, they proved the global existence and uniqueness of classical solution of the initial boundary problem with spatial dimension N 3 under the conditions that the coefficient of the cubic power term is negative or positive constant constant but initial energy E(u0) is sufficiently small. They also discussed the Blow-up property of classical solutions when the coefficient of the cubic power term is negative constant and initial energy E(u0) is sufficiently large.Because of the existence of the attractor can show the stability of the solution in some sense, the studies about the attractor of the Cahn-Hilliard are interesting. Dlotko, T. [11] considered solutions of the Cahn-Hilliard equation ut = - 2 2u + f(u), in Rn, n...