EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY

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1 Electronic Journal of Differential Equations, Vol ), No. 329, pp ISSN: URL: or EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY XIAOLI ZHANG, CHANGCHUN LIU Abstract. This article we study the Cahn-Hilliard/Allen-Cahn equation with degenerate mobility. Under suitable assumptions on the degenerate mobility and the double well potential, we prove existence of weak solutions, which can be obtained by considering the limits of Cahn-Hilliard/Allen-Cahn equations with non-degenerate mobility. 1. Introduction In this article, we consider a scalar Cahn-Hilliard/Allen-Cahn equation with degenerate mobility u t = [Du) u fu))] + u fu)), in, 1.1) where =, T ), is a bounded domain in R n with a C 3 -boundary and fu) is the derivative of a double-well potential F u) with wells ±1. The mobility Du) CR; [, )) is in the form Du) = u m, if u < δ, C Du) C 1 u m, if u δ, 1.2) for some constants C, C 1, δ >, where < m < if n = 1, 2 and 4 n < m < 4 if n 3. Equation 1.1) is supplemented by the boundary conditions and the initial condition u = u =, t >, 1.3) ux, ) = u x). 1.4) Equation 1.1) was introduced as a simplification of multiple microscopic mechanisms model [8] in cluster interface evolution. Equation 1.1) with constant mobility has been intensively studied. Karali and Nagase [9] investigated existence of weak solution to 1.1) with Du) D and a quartic bistable potential F u) = 1 u 2 ) 2. Karali and Nagase [9] only provided existence of the solution for the deterministic case. Then Antonopoulou, Karali and Millet [2] studied the stochastic case. The 21 Mathematics Subject Classification. 35G25, 35K55, 35K65. Key words and phrases. Cahn-Hilliard/Allen-Cahn equation; existence; Galerkin method; degenerate mobility. c 216 Texas State University. Submitted September 2, 216. Published December 24,

2 2 X. ZHANG, C. LIU EJDE-216/329 main result of this paper is the existence of a global solution, under a specific sublinear growth condition for the diffusion coefficient. Path regularity in time and in space is also studied. In addition, Karali and Ricciardi [7] constructed special sequences of solutions to a fourth order nonlinear parabolic equation of the Cahn- Hilliard/Allen-Cahn equation, converging to the second order Allen-Cahn equation. They studied the equivalence of the fourth order equation with a system of two second order elliptic equations. Karali and Katsoulakis [8] focus on a mean field partial differential equation, which contains qualitatively microscopic information on particle-particle interactions and multiple particle dynamics, and rigorously derive the macroscopic cluster evolution laws and transport structure. They show that the motion by mean curvature is given by V = µσκ, where κ is the mean curvature, σ is the surface tension and µ is an effective mobility that depends on the presence of the multiple mechanisms and speeds up the cluster evolution. This is in contrast with the Allen-Cahn equation where the velocity equals the mean curvature. Tang, Liu and Zhao [18] proved the existence of global attractor. Liu and Tang [15] obtained the existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. During the past few years, many authors have paid much attention to the Cahn- Hilliard equation with degenerate mobility. An existence result for the Cahn- Hilliard equation with a degenerate mobility in a one-dimensional situation has been established by Yin [19]. Elliott and Garcke [5] considered the Cahn-Hilliard equation with non-constant mobility for arbitrary space dimensions. Based on Galerkin approximation, they proved the global existence of weak solutions. Dai and Du [4] improved the results of the paper [5]. Liu [12] proved the existence of weak solutions for the convective Cahn-Hilliard equation with degenerate mobility. The relevant equations or inequalities have also been studied in [1, 11, 13, 14]. Motivated by the above works, we prove the existence of weak solution to 1.1)- 1.4) under a more general range of the double-well potential F. In particular, we assume that for s R, F C 2 R) satisfies k s r+1 1) F s) k 1 s r+1 + 1), 1.5) F s) k 2 s r + 1), 1.6) F s) k 3 s r 1 + 1), 1.7) for some constants k, k 1, k 2, k 3 > where 1 r < if n = 1, 2 and 1 r if n 3. What s more, we need the assumption on the boundary of fu), n fu) =, t >. 1.8) We can give examples satisfying the condition 1.8), such as F u) = 1 u 2 ) 2 studied by Karali and Nagase [9], the logarithmic function fu) = θ c u + θ 1+u 2 ln 1 u, u 1, 1), < θ < θ c [3]. Concerning the Allen-Cahn structure, we rewrite 1.1), 1.3), 1.4) and 1.8) to the form u t = Du) v) v, in, v = u + fu), in, ux, ) = u x), in, u = v =, on. 1.9)

3 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 3 We consider the free energy functional Eu) defined in [9] given by 1 Eu) := 2 u 2 + F u)) dx. 1.1) For a pair of solution u, v) of 1.9) it holds that d dt Eu) = vu t dx = v[ Du) v) v] dx = Du) v 2 + v 2) dx. Notation. Define the usual Lebesgue norms and the L 2 -inner-product u p = u Lp ) and u, v) = u, v) L2 ). The duality pairing between the space H 2 ) and its dual H 2 )) will be denoted using the form,. For simplicity, 2 := 2n. χ B denotes the characteristic function of B. This paper is organized as follows. In Section 2, we use a Galerkin method to give a existence of weak solution for a positive mobility. Section 3 uses a sequence of non-degenerate solutions to approximate the degenerate case 1.9). 2. Existence for positive mobility In this section, we study the Cahn-Hilliard/Allen-Cahn equation with a nondegenerate mobility D ε u) defined for an ε satisfying < ε < δ by { u m, if u > ε, D ε u) := ε m 2.1), if u ε. So we consider the problem u t = D ε u) v) v, in, v = u + fu), in, ux, ) = u x), in, u = v =, on. 2.2) Theorem 2.1. Suppose u H 1 ), under assumptions 1.2) and 1.5) 1.7), for any T >, there exists a pair of functions u ε, v ε ) such that 1) u ε L, T ; H 1 )) C[, T ]; L p )) L 2, T ; H 3 )), where 1 p < if n = 1, 2 and 2 p < 2n if n 3, 2) t u ε L 2, T ; H 2 )) ), 3) u ε x, ) = u x) for all x, 4) v ε L 2, T ; H 1 )), which satisfies equation 2.2) in the following weak sense T t u ε, φ dt + uε + fu ε ) ) φ dx dt Q T = D ε u ε ) 2.3) u ε + F ) u ε ) u ε φ dx dt for all test functions φ L 2, T ; H 2 ) H 1 )). In addition, u ε satisfies the energy inequality t Eu ε ) + Dε u ε x, τ)) v ε x, τ) 2 + v ε x, τ) 2) dx dτ Eu ), 2.4)

4 4 X. ZHANG, C. LIU EJDE-216/329 for all t >. To prove the above theorem, we apply a Galerkin approximation. Let {φ J } j N be the eigenfunctions of the Laplace operator on L 2 ) with Dirichlet boundary condition, i.e., φ J = λ J φ J, in, 2.5) φ J =, on. The eigenfunctions {φ J } j=1 form an orthogonal basis for L2 ), H 1 ) and H 2 ). Hence, for initial data u H 1 ), we can find sequences of scalars u N,j ; j = 1, 2,..., N) N=1 such that N lim u N,jφ J = u, in H 1 ). 2.6) N j=1 Let V N denote the linear span of φ 1,..., φ N ) and P N be the orthogonal projection from L 2 ) to V N, that is N ) P N φ := φφ J dx φ J. j=1 Let u N x, t) = N j=1 cn J t)φ Jx), v N x, t) = N j=1 dn J t)φ Jx) be the approximate solution of 2.2) in V N ; that is, u N, v N satisfy the g system of equations t u N φ J dx = D ε u N ) v N φ J dx v N φ J dx, 2.7) v N φ J dx = u N φ J + fu N )φ J dx, 2.8) u N x, ) = N u N,jφ J x), 2.9) j=1 for j = 1,..., N and u N,j = u φ J dx. This gives an initial value problem for a system of ordinary differential equations for c 1,..., c N ) N t c N J t) = d N N k t) D ε c N i t)φ i x) ) φ k φ J dx d N J t), 2.1) k=1 i=1 d N J t) = λ J c N J t) + f N c N i t)φ i x) ) φ J dx, 2.11) i=1 c N J ) = u N,j = u, φ J ), 2.12) which has to hold for j = 1,..., N. Define Xt) = c N 1 t),..., c N N t)), Ft, Xt)) = f 1 t, Xt)),..., f N t, Xt)) ), where N N ) f J t, Xt)) = D ε c N i t)φ i x) φ k φ J dx k=1 λ k c N k t) + i=1 f N i=1 c N i t)φ i x) ) ) φ k dx

5 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 5 λ J c N J t) f N k=1 ) c N k t)φ k x) φ J dx for j = 1,..., N. Then problem 2.1)-2.12) is equivalent to the problem X t) = Ft, Xt)), X) = u N,1,..., u N,N). Since the right hand side of the above equation is continuous, it follows from the Cauchy-Peano Theorem [16] that the problem 2.1)-2.12) has a solution Xt) C 1 [, T N ], for some T N >, i. e., the system 2.7)-2.9) has a local solution. To prove the existence of solutions, we need some a priori estimates on u N. Lemma 2.2. For any T >, we have where C independent of N. u N L,T ;H 1 )) C, for all N, t u N L 2,T ;H 2 )) ) C, for all N, Proof. For any fixed N N +, we multiply 2.7) by d N J t) and sum over j = 1,..., N to obtain t u N v N dx = D ε u N ) v N 2 dx v N 2 dx. 2.13) Multiply 2.8) by t c N J t) and sum over j = 1,..., N to obtain v N t u N dx = u N t u N + fu N ) t u N) dx, = d 1 dt 2 un 2 + F u N ) ) dx. By 2.13) and the above identity, we have d 1 dt 2 un 2 + F u N ) ) dx = D ε u N ) v N 2 dx v N 2 dx. 2.14) Replacing t by τ in 2.14) and integrating over τ [, t], by 1.5) and the Sobolev embedding theorem we obtain ) 1 2 un x, t) 2 + F u N x, t)) dx t + Dε u N x, τ)) v N x, τ) 2 + v N x, τ) 2) dx dτ ) 1 = 2 un x, ) 2 + F u N x, )) dx 1 2 un x, ) k 1 u N x, ) r+1 r+1 + k u k 1 C u r+1 H 1 ) + k 1 C. The last inequality follows from u H 1 ). This implies 1 2 un x, t) 2 + k u N r+1) dx t + Dε u N x, τ)) v N x, τ) 2 + v N x, τ) 2) dx dτ C. 2.15)

6 6 X. ZHANG, C. LIU EJDE-216/329 By 2.15) and Poincaré s inequality we have u N H 1 ) C, for t >. This estimate implies that the coefficients {c N J : j = 1,..., N} are bounded in time and therefore a global solution to the system 2.7)-2.9) exists. In addition, for any T >, we have u N L, T ; H 1 )), u N L,T ;H 1 )) C, for all N. 2.16) Inequality 2.15) implies D ε u N ) v N L 2 ) C, for all N, 2.17) v N L2 ) C, for all N. 2.18) By the Sobolev embedding theorem, the growth condition 1.2) and 2.1), for u > ε, we obtain D ε u N ) n/2 dx C 1 + 1) u N m n 2 dx, C u N mn/2 H 1 ) C. If u ε, obviously we obtain the above estimate. This implies D ε u N ) L,T ;L n/2 )) C, for all N. 2.19) For any φ L 2, T ; H 2 )), we obtain P N φ = N j=1 a Jt)φ J, where a J t) = φφ Jdx. Multiplying 2.7) by a J t), summing over j = 1, 2,..., N, by Hölder s inequality, 2.17)-2.19) and the Sobolev embedding theorem, we have T t u N φ dx dt T = = T T T t u N P N φ dx dt Dε u N ) v N P N φ + v N P N φ ) dx dt D ε u N ) n D ε u N ) v N 2 P N φ 2 dt + T C D ε u N ) v N 2 φ H 2 + v N 2 φ H 2 dt C D ε u N ) v N L 2 ) + v N L 2 )) φ L 2,T ;H 2 )) C φ L 2,T ;H 2 )). v N 2 P N φ 2 dt Hence, t u N L 2,T ;H 2 )) ) C for all N. 2.2) The proof is complete. Lemma 2.3. Suppose u H 1 ), under assumptions 1.2) and 1.5)-1.7), for any T >, there exists a pair of functions u ε, v ε ) such that 1) u ε L, T ; H 1 )) C[, T ]; L p )), where 1 p < if n = 1, 2 and 2 p < 2n if n 3, 2) t u ε L 2, T ; H 2 )) ), 3) u ε x, ) = u x) for all x,

7 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 7 4) v ε L 2, T ; H 1 )), which satisfies T t u ε, φ dt = T T D ε u ε ) v ε φ dx dt v ε φ dx dt. Proof. Since the embedding H 1 ) L p ) is compact for 1 p < if n = 1, 2 and 1 p < 2n if n 3, Lp ) H 2 )) is continuous for p 1 if n 3, p > 1 if n = 4 and p 2n n+4 if n 5. Using the Aubin-Lions lemma Lions [17]), we can find a subsequence which we still denote by u N and u ε L, T ; H 1 )), such that as N u N u ε, weak-* in L, T ; H 1 )), 2.21) u N u ε, strongly in C[, T ]; L p )), 2.22) u N u ε, strongly in L 2, T ; L p )) and almost everywehre in, 2.23) t u N t u ε, weakly in L 2, T ; H 2 )) ), 2.24) where 2 p < 2 if n 3 and 1 p < if n = 1, 2. By multiplying 2.7) by a J t) and integrating 2.7) over t [, T ], we obtain T t u N a J t)φ J dx dt T T 2.25) = D ε u N ) v N a J t) φ J dx dt v N a J t)φ J dx dt. To pass to the limit in 2.25), we need the convergence of v N and D ε u N ) v N. By 2.17) and D ε u N ) ε m, then v N L 2 ) Cε m 2 <, for any ε >. 2.26) This implies that { v N } is a bounded sequence in L 2 ), thus there exists a subsequence, not relabeled, and ζ ε L 2 ) such that By 2.26) and Poincaré s inequality, we have v N ζ ε, weakly in L 2 ). 2.27) v N L2,T ;H 1 )) Cε m 2 <, for any ε >. Hence we can find a subsequence of v N, not relabeled, and v ε L 2, T ; H 1 )) such that v N v ε, weakly in L 2, T ; H 1 )). 2.28) For any g L 2, T ; H 1 )), by 2.26) and 2.27) we have T T v N g dx dt = ζ ε g dx dt lim N = lim N T Hence ζ ε = v ε almost all in and v N g dx dt = T v ε g dx dt. v N v ε, weakly in L 2 ). 2.29)

8 8 X. ZHANG, C. LIU EJDE-216/329 By 2.18), we can extract a further sequence of v N, not relabeled, and η ε L 2 ) such that v N η ε, weakly in L 2 ). 2.3) By 2.28) and 2.3) for any g L 2 ) L 2, T ; H 1 )), we have T T T v N g dx dt = v ε g dx dt = η ε g dx dt. lim N This implies η ε = v ε almost all and Consequently we have the bound v N v ε, weakly in L 2 ). 2.31) v ε 2 dx dt C. 2.32) For any t [, T ], by D ε u N ) C1 + u N m ), we have Dε u N ) ) n/2 C1 + u N m ) n/2 C1 + u N )) mn/2, where 2 mn 2 < 2. By 2.22), C1 + u N ) C1 + u θ ) in L mn/2 ). Since D ε is continuous and 2.23), we obtain D ε u N ) D ε u ε ), a.e. in. The generalized Lebesgue convergence theorem [1] gives This implies D ε u N ) D ε u ε ), in L n/2 ). D ε u N ) D ε u ε ) n/2, as N. The above estimate holds for any t [, T ], and we can take supremum on both sides of the above estimate to obtain This implies sup D ε u N ) D ε u ε ) n/2, as N. t [,T ] D ε u N ) D ε u ε ), strongly in C, T ; L n/2 )). 2.33) By D ε u N ) C1 + u N m 2 ), 2.22), 2.23) and the generalized Lebesgue convergence theorem, similarly, we have D ε u N ) D ε u ε ), strongly in C, T ; L n )). 2.34) For any ϕ L 2, T ; L 2 )), by Hölder s inequality we have D ε u N ) v N ϕ ) D ε u ε ) v ε ϕ dx dt = [ D ε u N ) D ε u ε )] v N ϕ + ) D ε u ε )[ v N ϕ v ε ϕ] dx dt T D ε u N ) D ε u ε ) n v N 2 ϕ 2 dt + Dε u ε )ϕ[ v N v ε ] dx dt

9 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 9 sup t [,T ] D ε u N ) D ε u ε ) n v N L 2 ) ϕ L2,T ;L 2 )) + Dε u ε )ϕ[ v N v ε ] dx dt I + II. By 2.29) and 2.34), I as N. By Hölder s inequality and 2.34) we have T D ε u ε )ϕ 2 dx dt Dε u ε ) ) n/2 n/2 ) dx) ϕ 2n n dx dt This implies sup D ε u ε ) 2 n t [,T ] C ϕ 2 L 2,T ;L 2 )). T ϕ 2 L 2 ) dt Dε u ε )ϕ L 2 ). 2.35) Thus II as N by 2.29). Hence D ε u N ) v N D ε u ε ) v ε, weakly in L 2, T ; L 2n n+2 )). 2.36) Next we consider the convergence of D ε u N ) v N. By 2.17), 2.36) and L 2 ) L 2, T ; L 2n n+2 )), we can extract a further sequence, not relabeled, such that D ε u N ) v N D ε u ε ) v ε, weakly in L 2 ). 2.37) By Hölder s inequality and 2.17), we have D ε u N ) v N D ε u ε ) v ε dx dt D ε u N ) v N L 2 ) D ε u ε ) v ε L 2 ) C D ε u ε ) v ε L 2 ), 2.38) where C is independent of ε. Taking the limit of 2.38) on both sides, by 2.37) we have D ε u ε ) v ε L2 ) C. 2.39) For any ϕ L 2, T ; L 2 )), by Hölder s inequality we obtain Dε u N ) v N ϕ D ε u ε ) v ε ϕ ) dx dt [ D ε u N ) D ε u ε )] D ε u N ) v N ϕ dx dt + Dε u ε )[ D ε u N ) v N ϕ D ε u ε ) v ε ϕ] dx dt T D ε u N ) D ε u ε ) n D ε u N ) v N 2 ϕ 2 dt + Dε u ε )ϕ[ D ε u N ) v N D ε u ε ) v ε ] dx dt

10 1 X. ZHANG, C. LIU EJDE-216/329 sup D ε u N ) D ε u ε ) n D ε u N ) v N L 2 ) ϕ L2,T ;L 2 )) t [,T ] + Dε u ε )ϕ[ D ε u N ) v N D ε u ε ) v ε ] dx dt = I + II. By 2.34) and 2.37), I as N. By 2.35) and 2.37), we have II as N. Thus D ε u N ) v N D ε u ε ) v ε, weakly in L 2, T ; L 2n n+2 )). 2.4) For any φ L 2, T ; H 2 ) H 1 )), we obtain P n φ = n j=1 a Jt)φ J, where a J t) = φφ Jdx, then P n φ converges strongly to φ in L 2, T ; H 2 H 1 )) and a J t) L 2, T ). For φ J H 2 ), by Sobolev embedding theorem, we obtain Thus a J t) φ J L 2, T ; L 2 ) and φ J 2 C φ J H1 ) C. a J t)φ J L 2, T ; H 2 H 1 )) L 2, T ; H 1 )). Taking the limit as N on both sides of 2.25), by 2.24), 2.4) and 2.28), we have T t u ε, a J t)φ J dt T T = D ε u ε ) v ε a J t) φ J dx dt v ε a J t)φ J dx dt, for all j N. Then we sum over j = 1, 2,..., n on both sides 2.41) to get T = t u ε, P n φ dt T T D ε u ε ) v ε P n φ dx dt v ε P n φ dx dt. Since P n φ converges strongly to φ in L 2, T ; H 2 )), thus as n, T T P n φ φ 2 2 dt P n φ φ 2 H dt 1 T P n φ φ 2 H2 dt. 2.41) 2.42) This implies that P n φ converges strongly to φ in L 2, T ; L 2 )). Thus we obtain P n φ φ, weakly in L 2, T ; H 2 ) H 1 )), 2.43) P n φ φ, weakly in L 2, T ; L 2 )). 2.44) By L 2, T ; H 1 )) L 2, T ; H 1 )), we take the limit as n on both sides 2.42), then obtain T T T t u ε, φ dt = D ε u ε ) v ε φ dx dt v ε φ dx dt. 2.45)

11 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 11 As for the initial value, by 2.9) as N, u N x, ) u x) in L 2 ). By 2.22), u ε x, ) = u x) in L 2 ). The proof is complete. Proof of Theorem 2.1. We need only to check that u ε L 2, T ; H 3 )), v ε = u ε + fu ε ) and v ε = u ε + F u ε ) u ε. First we consider the convergence of u N and fu N ). By 2.21), we have T u N 2 2dt C. Hence we can find a subsequence of u N, not relabeled, and υ L 2 ), such that u N υ weakly in L 2 ). 2.46) For any φ L 2, T ; H 1 )), by integration by parts we have T T lim u N φ dx dt = lim u N φ dx dt. N N By 2.21), 2.46) and φ L 2 ) L 1, T ; H 1 )) we have T T T υφ dx dt = u ε φ dx dt = u ε φ dx dt. Hence υ = u ε almost all in [, T ] and u N u ε weakly in L 2 ). 2.47) By F u N ) C1 + u N r ), 2.22), 2.23) and the general dominated convergence theorem, similarly, we have F u N ) F u ε ) strongly in C, T ; L q )), 2.48) 2n r) for 1 q < if n = 1, 2 and 2 q < if n 3. By the growth condition 1.6) and the Sobolev embedding theorem, we obtain fu N ) 2 L 2 ) = F u N )) 2 dx C u N r + 1) 2 dx 2C u N 2r dx + 2C C u N 2r H 1 ) + C. Thus there exists a w L, T ; L 2 )) such that F u N ) w weakly-* in L, T ; L 2 )). This implies T T lim F u N )g dx dt = wg dx dt, 2.49) N for any g L 1, T ; L 2 )). By Hölder s inequality, 2.48) and 2.49), we have as N F u ε ) w ) g dx dt

12 12 X. ZHANG, C. LIU EJDE-216/329 F u ε ) F u N ) g dx dt + [F u N ) w]g dx dt T F u ε ) F u N ) 2 g 2 dt + [F u N ) w]g dx dt, for any g L 1, T ; L 2 )). Hence F u ε ) = w a.e. in and F u N ) F u ε ) weak-* in L, T ; L 2 )). 2.5) Multiplying 2.8) by a J t) and integrating 2.8) over t [, T ], we obtain T v N a J t)φ J dx dt T = un a J t) φ J + F u N )a J t)φ J ) dx dt. 2.51) For any φ L 2, T ; H 1 )), we obtain P n φ = n j=1 a Jt)φ J, where a J t) L 2, T ). Thus a J t)φ J L 2, T ; H 1 )) and a J t) φ J L 2 ). By 2.28), 2.47) and 2.5), we take the limit as N on both sides of 2.51) to get T T v ε a J t)φ J dx dt = u ε a J t) φ J + F u ε )a J t)φ J ) dx dt, 2.52) for all j N. Then we sum over j = 1,..., n on both sides 2.52), and obtain T T v ε P n φ dx dt = u ε P n φ + F u ε )P n φ) dx dt. 2.53) Since P n φ converges strongly to φ in L 2, T ; H 1 )), we have as n T P n φ φ 2 2 dt T P n φ φ 2 H dt. 1 This implies that P n φ converges strongly to φ in L 2 ). Thus we obtain P n φ φ weakly in L 2, T ; H 1 )) 2.54) P n φ φ weakly in L 2 ). 2.55) By L 2, T ; H 1 )) L 2, T ; H 1 )) and L, T ; L 2 )) L 2, T ; H 1 )), we take the limit as n on both sides 2.53), and we obtain v ε φ dx dt = u ε φ + F u ε )φ) dx dt. Since F u ε ) L, T ; L 2 )) and v ε L 2, T ; H 1 )), it follows from regularity theory [6] that u ε L 2, T ; H 2 )). Hence v ε = u ε + F u ε ) almost everywhere in. 2.56) Next we show F u ε ) L 2, T ; H 1 )). By Hölder s inequality, the Sobolev embedding theorem and 1.7), we have T T F u ε ) 2 dx dt = F u ε ) 2 u ε 2 dx dt T F u ε ) 2 ) n 2/n 2 dx u ε 2 ) n dx n dt

13 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 13 T C C C T T 1 + u ε r 1 ) n dx ) 2/n uε 2 2n dt uε 4 u ε r 1)n dx ) 2/n uε 2 H 1 ) dt 2n ) uε 2 H 2 ) dt C 1 + u ε 4 ) T L,T ;H 1 )) u ε 2 H 2 ) dt C 1 + u ε 4 L,T ;H 1 ))) uε 2 L 2,T ;H 2 )) C. Thus F u ε ) L 2 ) and F u ε ) L 2, T ; H 1 )). Combined with v ε L 2, T ; H 1 )), by 2.56) and regularity theory we have u ε L 2, T ; H 3 )) and v ε = u ε + F u ε ) u ε, almost everywhere in. 2.57) By 2.45), 2.56) and 2.57), we obtain T T t u ε, φ dt + u ε + F u ε ))φ dx dt = T D ε u ε ) u ε + F u ε ) u ε ) φ dx dt, 2.58) for all φ L 2, T ; H 2 ) H 1 )). Last we show that a weak solution u ε to 2.2) satisfies energy inequality 2.4). Replacing t by τ in 2.14) and integrating over τ [, T ], we have t Eu N x, t)) + D ε u N x, τ)) v N x, τ) 2 dx dτ t 2.59) + v N x, τ) 2 dx dτ = Eu N x, )). Next, we pass to the limit in 2.59). First, by mean value theorem and 1.6) we have F u N t)) F u ε t)) ) dx F ξ) u N t) u ε t) dx C u N t) r + u ε t) r + 1) u N t) u ε t) dx, 2.6) for 1 r < if n = 1, 2 and 1 r n if n 3, ξ = λun t) + 1 λ)u ε t) for some λ, 1). By Hölder s inequality, we have u N t) r u N t) u ε t) dx u N t) u ε t) 2 u N t) r 2r. 2.61) Since the Sobolev embedding theorem says that H 1 ) L p ) for 1 p 2 and the embedding is compact if 1 p < 2, by 2.21), then for a subsequence, not relabeled, we have u N u ε strongly in L, T ; L 2 )) and u N is bounded in L, T ; L 2r )). Hence, it follows from 2.61) that u N t) r u N t) u ε t) dx, 2.62)

14 14 X. ZHANG, C. LIU EJDE-216/329 as N, for almost all t [, T ]. Similarly, we can prove that u ε t) r + 1) u N t) u ε t) dx, 2.63) as N, for almost all t [, T ], by 2.6), 2.62) and 2.63), we have F u N t)) dx = F u ε t)) dx. 2.64) lim N Since u N x, ) u x) strongly in L 2 ), we obtain F u N )) dx = F u x)) dx. 2.65) lim N By 2.47), 2.64), 2.37), 2.29), 2.59) and the weak lower semicontinuity of the L p norms [3]. Then 1 2 u εx, t) 2 + F u ε x, t))) dx t + lim inf N + lim N inf Dε u ε x, τ)) v ε x, τ) 2 + v ε x, τ) 2) dx dτ ) 1 2 un x, t) 2 + F u N x, t)) dx = lim N inf Eu N x, )). Q t Dε u N x, τ)) v N x, τ) 2 + v N x, τ) 2) dx dτ 2.66) Since u N x, ) u x) strongly in H 1 ), by 2.65) we have 1 lim N EuN x, )) = 2 u x) 2 + F u x))) dx. 2.67) Combining 2.66) with 2.67) gives the energy inequality 2.4). The proof is complete. 3. Degenerate mobility This section is devoted to the existence of weak solutions to the equations 1.9). Here we consider the limit of approximate solutions u εi defined in section 2. The limiting value u does exist and solves the degenerate Allen-Cahn/Cahn-Hilliard equation in the weak sense. Theorem 3.1. Suppose u H 1 ), under assumptions 1.2) and 1.5)-1.7), for any T >, problem 1.9) has a weak solution u : R satisfying 1) u L, T ; H 1 )) C[, T ]; L p )) L 2, T ; H 2 )), where 1 p < if n = 1, 2 and 2 p < 2n if n 3, 2) t u L 2, T ; H 2 )) ), 3) ux, ) = u x) for all x, which satisfies 1.9) in the following weak sense:

15 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 15 1) Define P as the set where Du) is non-degenerate, that is P := {x, t) : u }. There exists a set A with \ A = and a function ζ : R n satisfying χ A P Du)ζ L 2, T ; L 2n n+2 )), such that T = t u, φ dt T A P T Du)ζ φ dx dt u + fu))φ dx dt 3.1) for all test functions φ L 2, T ; H 2 ) H 1 )). 2) For each j N, there exists E J := {x, t) ; u i u uniformly, u > δ J for δ J > } = T J S J such that u L 2 T J ; H 3 S J )), ζ = u + F u) u, in E J. In addition, u satisfies the energy inequality Eu) + Dux, τ)) ζx, τ) 2 dx dτ Q t A P + u + fu) 2 dx dτ Eu ), Q t for all t >. 3.2) Proof. We consider a sequence of positive numbers ε i monotonically decreasing to as i. Fix u H 1 ), for any fixed ε i, here, for the sake of simplicity, we write u i := u εi and D i u i ) := D εi u εi ). By Theorem 2.1, there exists a function u i such that 1) u i L, T ; H 1 )) C[, T ]; L p )) L 2, T ; H 3 )), where 1 p < if n = 1, 2 and 2 p < 2n if n 3, 2) t u i L 2, T ; H 2 )) ), T T T t u i, φ dt = D i u i ) v i φ dx dt v i φ dx dt 3.3) for all test functions φ L 2, T ; H 2 ) H 1 )), where v i = u i + fu i ), almost everywhere in. 3.4) By the arguments in the proof of Theorem 2.1, the bounds on the right hand side of 2.16), 2.2), 2.39) and 2.32) depend only on the growth conditions of the mobility and potential, so there exists a constant C > independent of ε i such that u i L,T ;H 1 )) C, 3.5) t u i L2,T ;H 2 )) ) C, 3.6) D i u i ) v i L2 ) C, 3.7) v i L2 ) C. 3.8)

16 16 X. ZHANG, C. LIU EJDE-216/329 Similar to the proof of Theorem 2.1, the above boundedness of {u i } and { t u i } enable us to find a subsequence, not relabeled, and u L, T ; H 1 )) such that as i, u i u, weak-* in L, T ; H 1 )), 3.9) u i u, strongly in C, T ; L p )), 3.1) u i u, strongly inl 2, T ; L p )) and almost all in, 3.11) t u i t u, weakly in L 2, T ; H 2 )) ), 3.12) where 1 p < if n = 1, 2 and 2 p < 2n if n 3. By 3.7) and 3.8), there exists ξ, η L 2 ) such that Di u i ) v i ξ, weakly in L 2 ), 3.13) v i η, weakly in L 2 ). 3.14) Next we show the convergence of D i u i ) v i and η = u + fu) a.e.. Similar to having 2.33) and 2.34), by the uniform convergence of D i D, we obtain D i u i ) Du), strongly in C, T ; L n/2 )), 3.15) Di u i ) Du), strongly in C, T ; L n )). 3.16) For any ϕ L 2, T ; L 2 )), by Hölder s inequality, we have D i u i ) v i ϕ ) Du)ξϕ dx dt [ D i u i ) Du)] D i u i ) v i ϕ dx dt + Du)[ Di u i ) v i ϕ ξϕ] dx dt T D i u i ) Du) n D i u i ) v i 2 ϕ 2 dt + Du)ϕ[ Di u i ) v i ξ] dx dt sup t [,T ] D i u i ) Du) n D i u i ) v i L 2 ) ϕ L2,T ;L 2 )) + Du)ϕ[ Di u i ) v i ξ] dx dt =: I + II. By 3.16) and 3.7), I as N. By Hölder s inequality and the boundedness of Du) in C, T ; L n/2 )) we have T Du)ϕ 2 ) n/2 n/2 ) dx dt Du) dx) ϕ 2 n dx dt T sup Du) n/2 ϕ 2 L 2 ) dt t [,T ] C ϕ 2 L 2,T ;L 2 )).

17 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 17 This implies Du)ϕ L 2 ). 3.17) By 3.13), thus II as N, this implies D i u i ) v i Du)ξ weakly in L 2, T ; L 2n n+2 )). 3.18) By 3.4), for any φ L 2, T ; H 1 )) L 2 ) we have v i φ dx dt = u i φ dx dt + fu i )φ dx dt Q T = u i φ dx dt + fu i )φ dx dt. 3.19) Recalling that the convergence of u i and fu i ) are similar to get 2.47) and 2.5), we have u i u, weak-* in L, T ; L 2 )), 3.2) fu i ) fu), weak-* in L, T ; L 2 )). 3.21) By 3.2), 3.21) and L 2, T ; H 1 )) L 1, T ; L 2 )), taking the limits of 3.19) on both sides, we have ηφ dx dt = u φ dx dt + fu)φ dx dt. Since fu) L, T ; L 2 )) and η L 2 ), by regularity theory we see that u L 2, T ; H 2 )) and η = u + fu), almost everywhere in. 3.22) By 3.12), 3.18) and 3.22), taking the limits of 3.3), we have T T T t u, φ dt = Du)ξ φ dx dt [ u + fu)]φ dx dt. 3.23) As for the initial value, since u i x, ) = u x) in L 2 ), by 3.1) we have ux, ) = u x). Now we consider the weak convergence of v i. Choose a sequence of positive numbers δ J that monotonically decreases to as j. By 3.11) and Egorov s theorem, for every δ J >, there exists a subset B J with \B J < δ J such that u i u, uniformly in B J. Define A 1 = B 1, A 2 = B 1 B 2,..., A J = B 1 B 2 B J. Then A 1 A 2 A J A j ) Thus the limit of {A J } exists, then we have lim j A J = j=1 A J := A and \A =. Define P J := {x, t) ; u > δ J }. Then P 1 P 2 P J P j ) Thus the limit of {P J } exists, then we have lim j P J = j=1 P J := P. For each j, we define E J := A J P J, G J := A J \P J, where u > δ J and u i u uniformly, where u δ j and u i u uniformly.

18 18 X. ZHANG, C. LIU EJDE-216/329 Thus we obtain A J = E J G J. By 3.24) and 3.25), we have E 1 E 2 E J E j+1. Thus the limit of {E J } exists, then we have lim j E J = j=1 E J = A P := E. For any ψ L 2, T ; L 2 )), D i u i ) v i ψ dx dt Q T = D i u i ) v i ψ dx dt + D i u i ) v i ψ dx dt 3.26) \A J G J + D i u i ) v i ψ dx dt. E J As i, by 3.18) we obtain lim D i u i ) v i ψ dx dt = Du)ξψ dx dt, 3.27) i Q T lim D i u i ) v i ψ dx dt = Du)ξψ dx dt. 3.28) i \A J \A J By \A =, taking the limit of 3.28) as j, we have lim lim D i u i ) v i ψ dx dt =. 3.29) j i \A J To analyze the second and third terms of 3.26), we write u j 1,i := u i and v j 1,i := v i in A J, then we have u j 1,i u, uniformly in A J for all j N. This implies that there exists an index N J N + such that for all i N J, u j 1,i u < δ J 2. We can easily get the following result: u j 1,i δ J 2, if x, t) E J, u j 1,i 2δ J, if x, t) G J. 3.3) Considering the limit of the second term of 3.26), by Hölder s inequality and 3.7) we have D j 1,i u j 1,i ) v j 1,i ψ dx dt G J D j 1,i u j 1,i ) D j 1,i u j 1,i ) v j 1,i ψ dx dt sup x,t) G J sup x,t) G J C sup x,t) G J D j 1,i u j 1,i ) D j 1,i u j 1,i ) v j 1,i L2 ) ψ L2 ) D j 1,i u j 1,i ) 1/n ψ L2,T ;L 2 )) C max{2δ J ) m/2, ε m/2 j 1,i }. 3.31)

19 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 19 Taking the limits of 3.31) as i and j, we have lim lim D j 1,i u j 1,i ) v j 1,i ψ dx dt j i G J 3.32) C lim lim max{2δ J) m/2, ε m/2 j 1,i } =. j i By 3.7) and 3.3), we obtain δ J 2 )m v j 1,i 2 dx dt D j 1,i u j 1,i ) v j 1,i 2 dx dt D J D J D j 1,i u j 1,i ) v j 1,i 2 dx dt C. This implies v j 1,i 2 dx dt Cδ J ) m. D J So v j 1,i is bounded in L 2 E J ), thus there exists a subsequence, labeled as { v j,i }, and ζ J L 2 E J ) such that v j,i ζ J, weakly in L 2 E J ). 3.33) By E j 1 E J, for any g L 2 E J ), we have g L 2 E j 1 ) and v j 1,i = v j,i in E j 1. By 3.33) we have lim v j,i g dx dt = lim v j 1,i g dx dt i E i j 1 E j 1 = ζ J g dx dt = ζ j 1 g dx dt. E j 1 E j 1 Thus ζ J = ζ j 1 almost everywhere in E j 1. we define { ζ J, if x, t) E J, ω J :=, if x, t) E\E J. So for almost every x, t) E, there exists a limit of ω J x, t) as j. We write ζx, t) = lim j ω Jx, t), almost everywhere in E. Clearly ζx, t) = ζ J x, t) for almost all x, t) E J for all j. Using a standard diagonal argument, we can extract a subsequence such that v k,nk ζ, weakly in L 2 E J ) for all j. 3.34) For any ϕ L 2, T ; L 2 )), by Hölder s inequality we have ) χ EJ D k,nk u k,nk ) v k,nk ϕ χ EJ Du)ζϕ dx dt χ EJ [ D k,nk u k,nk ) Du)] v k,nk ϕ dx dt + χ EJ Du)[ vk,nk ϕ ζϕ] dx dt sup D k,nk u k,nk ) T Du) n χ EJ v k,nk 2 ϕ 2n dt t [,T ]

20 2 X. ZHANG, C. LIU EJDE-216/329 + Du)ϕ[ vk,nk ζ] dx dt E J sup D k,nk u k,nk ) Du) n v k,nk L2 E J ) ϕ L 2,T ;L 2 )) t [,T ] + Du)ϕ[ vk,nk ζ] dx dt E J =: I + II. By 3.16) and 3.34), I as N. By 3.17) and 3.34), we have II as N. Thus χ EJ D k,nk u k,nk ) v k,nk χ EJ Du)ζ, weakly in L 2, T ; L 2n n+2 )), for all j. From L 2 L 2n n+2 and 3.13), we see that ξ = Du)ζ in every EJ and Consequently, by 3.18), ξ = Du)ζ in E. 3.35) χ E D k,nk u k,nk ) v k,nk χ E Du)ζ, weakly in L 2, T ; L 2n n+2 )). Thus by Taking the limits of third term of 3.26), we have lim lim D k,nk u k,nk ) v k,nk ψ dx dt j k E J = lim Du)ζψ dx dt = Du)ζψ dx dt. j E J E By 3.27), 3.29), 3.32) and 3.36), we have Du)ξψ dx dt = E 3.36) Du)ζψ dx dt. 3.37) By 3.23) and 3.37), we find that u and ζ solve equation 1.9) in the following weak sense T T t u, φ dt = Du)ζ φ dx dt [ u + fu)]φ dx dt, 3.38) E for all φ L 2, T ; H 2 ) H 1 )). From 3.14) and 3.34), we notice that v i is bounded in L 2 T J ; H 1 S J )), where E J = T J S J. So we can extract a further sequence, not relabeled, and v L 2 T J ; H 1 S J )), v i v weakly in L 2 T J ; H 1 S J )). 3.39) Similar to show F u ε ) L 2, T ; H 1 )) and 3.22). Hence, we have F u) L 2, T ; H 1 )) and v = u + fu), a.e. in E J, By v L 2 T J ; H 1 S J )) we have u L 2 T J ; H 3 S J )) and v = u + F u) u, almost everywhere in E J. 3.4) Obviously we have η = v, ζ = v, a. e. ine J. So we obtain the desired relation between ζ and u: ζ = u + F u) u, in E J.

21 EJDE-216/329 CAHN-HILLIARD/ALLEN-CAHN EQUATION 21 Finally, we show that a weak solution u to 1.9) satisfies energy inequality 3.2). By 2.4) we have 1 2 u k,n k x, t) 2 + F u k,nk x, t))) dx + D k,nk u k,nk x, τ)) v k,nk x, τ) 2 dx dτ Q t E 3.41) + v k,nk x, τ) 2 dx dτ Q t 1 2 u 2 + F u )) dx. By having 2.47) and 2.66), similarly we have u k,nk u, weakly in L 2 ), 3.42) F u k,nk t)) dx = F ut)) dx. 3.43) lim N By 3.42), 3.43), 3.13), 3.35), 3.14), 3.22), 3.41) and the weak lower semicontinuity of the L p norms. Then ) 1 2 ux, t) 2 + F ux, t)) dx + Dux, τ)) ζx, τ) 2 dxdτ Q t E + u + fu) 2 dx dτ Q t 1 lim inf N 2 u k,n k x, t) 2 + F u k,nk x, t))) dx + lim inf D k,nk u k,nk x, τ)) v k,nk x, τ) 2 dx dτ N Q t E + lim inf v k,nk x, τ) 2 dxdτ N Q t 1 2 u 2 + F u )) dx. This gives the energy inequality 3.2). The proof is complete. Acknowledgments. The authors would like to thanks the anonymous referees for the valuable suggestions for the revision and improvement of this article. References [1] H. W. Alt; Lineare Funktionalanalysis, Springer-Verlag, Berlin, [2] D. C. Antonopoulou, G. Karali, A. Millet; Existence and regularity of solution for a stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion, J. Differential Equations, 263) 216), [3] L. Cherfils, A. Miranville, S. Zelik; The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 792) 211), [4] S. Dai, Q. Du; Weak solutions for the Cahn-Hilliard equation with degenerate mobility, Arch. Rational Mech. Anal., ), [5] C. M. Elliott, H. Garcke; On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 272) 1996), [6] L. C. Evans; Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.

22 22 X. ZHANG, C. LIU EJDE-216/329 [7] G. Karali, T. Ricciardi; On the convergence of a fourth order evolution equation to the Allen-Cahn equation, Nonlinear Analysis, 72 21), [8] G. Karali, M. A. Katsoulakis; The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 23527), [9] G. Karali, Y. Nagase; On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation, Discrete and Continuous Dynamical Systems Series S, 71) 214), [1] B. Liang; Existence and asymptotic behavior of solutions to a thin film equation with Dirichlet boundary, Nonlinear Analysis: Real World Applications, ), [11] B. Liang, R. Ji, Y. Zhu; Positive solutions to a nonlinear fourth-order partial differential equation, Nonlinear Analysis: Real World Applications, ), [12] C. Liu; On the Convective Cahn-Hilliard equation with degenerate mobility, Journal of Mathematical Analysis and Applications, 3441) 28), [13] C. Liu; A fourth order nonlinear degenerate parabolic equation, Communications on Pure and Applied Analysis, 73) 28), [14] C. Liu; A sixth-order thin film equation in two space dimensions, Advances in Differential Equations, 25/6) 215), [15] C. Liu, H. Tang; Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions, Accepted by Evolution Equations and Control Theory. [16] T. C. Sideris; Ordinary Differential equations and Dynamical Systems, Atlantis Studies in Differential Equations, Atlantis Press, Paris, 213. [17] J. Simon; Compact sets in the space L p, T ; B), Ann. Mat. Pura Appl., ), [18] H. Tang, C. Liu, Z. Zhao; The existence of global attractor for a Cahn-Hilliard/Allen-Cahn equation, Bulletin of the Iranian Mathematical Society, 423) 216), [19] J. Yin; On the existence of nonnegative continuous solutions of the Cahn-Hilliard equation, J. Differential Equations, ), Xiaoli Zhang Department of Mathematics, Jilin University, Changchun 1312, China address: mathche@sina.com Changchun Liu corresponding author) Department of Mathematics, Jilin University, Changchun 1312, China address: liucc@jlu.edu.cn