Weak Solutions for a Sixth Order Cahn-Hilliard Type Equation with Degenerate Mobility

and Applied Analysis 3 By use of the Peano existence theorem, the initial value problem has a local solution. Now, we set

Equation ( 1) is the sixth order parabolic equation which describes dynamics of phase transitions in ternary oil-watersurfactant systems [2].Here (, ) is the scalar order parameter which is proportional to the local difference between oil and water concentrations.Pawłow and Zajączkowski [2] proved that the initial-boundary value problem (1)-( 4) with () = 1 admits a unique global smooth solution which depends continuously on the initial datum.Wang and Liu [3] proved that the solutions of problem ( 1)- (4) with () = 1 might not be classical globally.In other words, in some cases, the classical solutions exist globally, while in some other cases, such solutions blow up at a finite time.They also discussed the existence of global attractor.Liu and Wang [4] considered the optimal control problem for the problem (1)-(4) with () = 1.They proved the existence of optimal solution.The optimality system is also established.Since the mobility depends on the concentration in general, the equation with nonlinear main part reflects even more exactly the physical reality comparing to the one with linear main part.Schimperna and Pawłow [5] studied (1) with viscous term Δ  , () = 1 and logarithmic potential They investigated the behavior of the solutions to the sixth order system as the parameter  tends to 0. The uniqueness and regularization properties of the solutions have been discussed.Liu [6] studied the problem (1)-( 4) and he proved the existence of classical solutions for two dimensions and

Existence for Positive Mobilities
In this section, we study the Cahn-Hilliard equation with a mobility which is bounded away from zero.We prove existence of weak solutions.Consider the following sixth order Cahn-Hilliard equation: with the boundary conditions and the initial value In this section, we assume Ω is a bounded domain with Lipschitz boundary and  such that (i) () ∈ (,  + ) and there is  1 > 0 such that  1 ≤ ().
Under these assumptions we can state the following theorem.
Proof.To prove the theorem we apply a Galerkin approximation.Let {  } ∈ be the eigenfunctions of the Laplace operator with Neumann boundary conditions; that is, The eigenfunction   is orthogonal in the  2 (Ω),  1 (Ω), and  2 (Ω) scalar product.We normalize the   such that (  ,   )  2 (Ω) =   .Furthermore we assume without loss of generality that  1 = 0. Now we consider the following Galerkin ansatz for (6): This gives an initial value problem for a system of ordinary differential equations for ( 1 ,  2 , . . .,   ): By use of the Peano existence theorem, the initial value problem has a local solution.Now, we set In order to derive a priori estimates we differentiate the () and get This implies The last inequality follows the fact that  0 ∈  2 (Ω).

Existence for the Degenerate Case
Our approach is to approximate the degenerate problem by nondegenerate equations.By Theorem 1, we know the existence of weak solution to the problem where   () = () + .
In the next step we prove the energy estimates.

Lemma 3.
There exists an  0 such that for all 0 <  <  0 , and the following estimates hold with a constant C independent of  (1) ess sup and (2) follows easily from (1), and this finishes the proof of Lemma 3.