Nonlinear diffusion equations with Robin boundary conditions as asymptotic limits of Cahn Hilliard systems

Transcription

1 arxiv:18.755v1 [math.ap] 8 Feb 18 Nonlinear diffusion equations with Robin boundary conditions as asymptotic limits of Cahn illiard systems Takeshi Fukao Department of Mathematics, Faculty of Education Kyoto University of Education 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto Japan fukao@kyokyo-u.ac.jp Taishi Motoda Graduate School of Education, Kyoto University of Education 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto Japan motoda.math@gmail.com Abstract Condition imposed on the nonlinear terms of a nonlinear diffusion equation with Robin boundary condition is the main focus of this paper. The degenerate parabolic equations, such as the Stefan problem, the ele Shaw problem, the porous medium equation and the fast diffusion equation, are included in this class. By characterizing this class of equations as an asymptotic limit of the Cahn illiard systems, the growth condition of the nonlinear term can be improved. In this paper, the existence and uniqueness of the solution are proved. From the physical view point, it is natural that, the Cahn illiard system is treated under the homogeneous Neumann boundary condition. Therefore, the Cahn illiard system subject to the Robin boundary condition looks like pointless. owever, at some level of approximation, it makes sense to characterize the nonlinear diffusion equations. Key words: Cahn illiard system, degenerate parabolic equation, Robin boundary condition, growth condition. AMS MOS) subject classification: 35K61, 35K65, 35K5, 35D3, 8A. 1

2 Nonlinear diffusion equations with Robin boundary conditions 1 Introduction We consider the initial boundary value problem of a nonlinear diffusion system P), comprising a parabolic partial differential equation with a Robin boundary condition: u ξ = g, ξ βu) in Q :=, T ), t P) ν ξ + κξ = h on Σ :=, T ), u) = u in, where κ is a positive constant. In an asymptotic form, it is characterized as the limit of the Cahn illiard system with a Robin boundary condition, u ε t µ ε = in Q, P) µ ε = ε u ε + ξ ε + π ε u ε ) f, ξ ε βu ε ) in Q, ε ν u ε + κu ε =, ν µ ε + κµ ε = on Σ, u ε ) = u ε in as ε with ξ := µ + f, where < T < +, is a bounded domain of R d d =, 3) with smooth boundary :=, ν denotes the outward normal derivative on, is the Laplacian. Functions g : Q R, h : Σ R, u : R and u ε : R are given as the boundary and initial data. f : Q R is constructed from g and h later. Moreover, in the nonlinear diffusion term, β is a maximal monotone graph and π ε is an anti-monotone function that tends to as ε. It is well known that the Cahn illiard system is characterized by the nonlinear term β + π ε, a simple example being βr) = r 3 and π ε r) = εr for all r R. In this way, we choose a suitable π ε that depends on the definition of β yielding the structure of the Cahn illiard system for P) ε. Alternatively, by choosing a suitable β, the degenerate parabolic equation P) characterizes various types of nonlinear problems, such as the Stefan problem, ele-shaw problem, porous medium equation, and fast diffusion equation see, e.g., [7, pp , Examples]). In analyzing the well-posedness of system P), there are two standard approaches, the L 1 -approach and the ilbert space approach see, e.g., [4], [5, Chapter 5]). With respect to the ilbert space approach, the pioneering result [9] concerns the enthalpy formulation for the Stefan problem with a Dirichelet Robin boundary condition, essentially of Robin-type. Afterwards, the Neumann boundary condition was treated in [15,18] and the dynamic boundary condition in [1 3, 14]. See also [13, 19, ] for a more general space setting. For all of these results, the growth condition for β is a very important assumption, such as βr) c 1 r c for all r R, where c 1 and c are positive constants, and β is a proper lower semicontinuous convex function satisfying the subdifferential form R β = β. owever, it is too restricted in regard to application; indeed, for fast diffusion or nonlinear diffusion of Penrose Fife type are excluded. A drawback of the ilbert space approach compared with the L 1 -approach is detailed in [5, Chapter 5]. With that as motivation, the improvement of the growth condition subject to the Robin boundary condition, was studied in [1]

3 Takeshi Fukao and Taishi Motoda 3 using a certain technique called the lower semicontinuous convex extension. For porous medium, that is, βr) = r q 1 r, q > 1), a different approach to the doubly nonlinear evolution equation was studied in [4]. Regarding a recent result, the characterization of the nonlinear diffusion equation as an asymptotic limit of the Cahn illiard system with dynamic boundary conditions was introduced in [11, 1], and the same problem subject to the Neumann boundary condition was given in [7]. In these instances, we do not need any growth condition; see [7, Chapter 6] or [1]. This is one of the big advantages of this approach; indeed, we do not need techniques such as the lower semicontinuous convex extension by [1]. The main objective of this paper is to improve on the pioneering result give in [9,1], more precisely, an improvement of the growth condition subject to the Robin boundary condition without using the lower semicontinuous convex extension. Up to a certain level of approximation, we consider the Cahn illiard system subject to the Robin boundary condition see, e.g., [1]). From a physical perspective, the Cahn illiard system is more naturally treated under the homogeneous Neumann boundary condition. Therefore the Cahn illiard system with the Robin boundary condition imposed looks to be without points. owever, up to a given level of approximation, characterizing the nonlinear diffusion equations with the Robin boundary condition imposed does make sense. Moreover, we obtain the order of convergence for the solutions of P) with the Robin boundary condition as that for P) with the Neumann boundary condition, that is, letting κ tend to, where κ is the constant in the boundary condition. The outline of the paper is as follows. In Section, the main theorems are stated. For this purpose, we present the notation used in this paper and define a suitable duality map and the 1 -norm equivalent to the standard norm. Next, we introduce the definition of a weak solution of P) and P) ε ; the principal theorems are then given. In Section 3, to prove the convergence theorem, we deduce the uniform estimates of the approximate solution of P) ε. We use Moreau Yosida regularization of β employing the second-order approximate of parameter λ. In Section 4, to obtain the weak solution of P) ε, we first pass to the limit λ. Second, we prove the existence of weak solutions by passing to the limit ε. We also discuss the uniqueness of solutions. In Section 5, we improve the assumption for β subject to a strong assumption for the heat source f. From this results, we can avoid the growth condition for β. In Section 6, we obtain the order of convergence related to the Neumann problem from the Robin problem as κ. Table of contents: 1. Introduction. Main results.1. Notation.. Definition of the solution and main theorem 3. Approximate problem and uniform estimates 3.1. Approximate problem for P) ε 3.. Uniform estimates

4 4 Nonlinear diffusion equations with Robin boundary conditions 4. Proof of convergence theorem 4.1. Passage to the limit λ 4.. Passage to the limit ε 5. Improvement of the results 6. Asymptotic limits to solutions of Neumann problem Appendix Main results In this section, we state the main theorem. ereafter, κ is a positive constant..1 Notation We employ spaces := L ) and := L ), with standard norms and, along with inner products, ) and, ), respectively. We also use the space V := 1 ) with norm V and inner product, ) V, z V := z d + κ z ) 1 for all z V. By virtue of the Poincaré inequality and the trace theorem, there exist positive constants c P, c P with c P < c P such that c P z V z V c P z V for all z V, where V stands for the standard 1 -norm. Moreover, we set W := { z ) : ν z + κz = a.e. on }. The symbol V denotes the dual space of V ; the duality pairing between V and V is denoted by, V,V. Also, for all z V, let F : V V be the duality mapping defined by F z, z V,V := z zdx + κ z zd for all z, z V. Moreover, we define an inner product of V by z, z ) V := z, F 1 z V,V for all z, z V. Then, the dense and compact embeddings V V hold; that is, V,, V ) is a standard ilbert triplet. As a remark, for all of these settings, κ > is essential. For a Neumann boundary condition, κ =, see [7].. Definition of the solution and main theorem We next define our solution for P) and then state the main theorem.

5 Takeshi Fukao and Taishi Motoda 5 Definition.1. The pair u, ξ) is called the weak solution of P) if and they satisfy u t), z V,V + u 1, T ; V ) L, T ; ), ξ L, T ; V ), ξ βu) a.e. in Q, ξt) zdx + κ ξt)zd = gt)zdx + ht)zd for all z V, for a.a. t, T ),.1) u) = u a.e. in..) In this definition, the Robin boundary condition for ξ is hidden in the weak formulation.1). The strategy behind the proof of the main theorem is the characterization of our nonlinear diffusion equation P) as an asymptotic limit of the Cahn illiard system. Therefore, for each ε, 1], we define the approximate problem of Cahn illiard type with Robin boundary condition as follows: Definition.. The triplet u ε, µ ε, ξ ε ) is called the weak solution of P) ε if u ε 1, T ; V ) L, T ; V ) L, T ; W ),.3) µ ε L, T ; V ), ξ ε L, T ; ), ξ ε βu ε ) a.e. in Q and they satisfy u ε t), z + µ V,V ε t) zdx + κ µ ε t)zd = for all z V,.4) µ ε t) = ε u ε t) + ξ ε t) + π ε uε t) ) ft) in,.5) for a.a. t, T ), with u ε ) = u ε a.e. in. The Robin boundary condition for u ε is stated with regard to the class of W, that is, the regularity.3); that for µ ε is hidden in the weak formulation.4). The Cahn illiard structure is characterized by the nonlinear term β + π ε. The conditions for these terms are given as assumptions: A1) β : R R is a maximal monotone graph, which is the subdifferential β = R β of some proper lower semicontinuous convex function β : R [, + ] satisfying β) = with the effective domain Dβ) := {r R : βr) }; A) there exist positive constants c 1, c such that βr) c 1 r c for all r R; A3) π ε : R R is a Lipschitz continuous function for all ε, 1]. Moreover, there exist a positive constant c 3 and strictly increasing continuous function σ : [, 1] [, 1] such that σ) =, σ1) = 1, and πε ) + π ε L R) c 3 σε) for all ε, 1]..6)

6 6 Nonlinear diffusion equations with Robin boundary conditions In particular, A1) yields β). Assumption A) is improved in Section 6. The assumptions pertaining to the given data are as follows: A4) g L, T ; ), h L, T ; ); A5) u with βu ) L 1 ). Moreover, let u ε V ; then there exists a positive constant c 4 such that, for all ε, 1], u ε c 4, βu ε )dx c 4, ε u ε c d 4, ε u ε c 4..7) In addition, u ε u strongly in as ε cf. [7, Lemma A.1]). From assumption A4), we see from the Lax Milgram theorem that there exists a unique function f L, T ; V ) such that ft) zdx + κ ft)zd = gt)zdx + ht)zd for all z V and for a.a. t, T ). Therefore, introducing new variable µ := ξ f L, T ; V ), we rewrite the weak formulation.1) as follows: u t), z + µt) zdx + κ µt)zd = for all z V,.8) V,V for a.a. t, T ). The proof of main theorem follows that in [7,11] for the Robin boundary condition. The characterization of the nonlinear diffusion equation from the asymptotic limits of Cahn illiard system [7, 11 13] furnishes a big advantage in regard to the growth condition for β. Because of this, we can improve the result [9] and widen the setting for β using the different approach described in [1] starting from the lower semicontinuous convex extension. To do so, we replace assumption A4) by the following A6): A6) g L, T ; ), h = a.e. on Σ. Then, we see that there exists a unique function f L, T ; V ) such that, ft) zdx + κ ft)zd = gt)zdx for all z V, for a.a. t, T ). Now, taking a test function z D), we have ft) = gt) in D ). Specifically, by comparison, ft) = gt) in. This yields ν ft) + κft) = a.e. on, for a.a. t, T ). Therefore, under assumption A6), we have f L, T ; ) and ν f L, T ; ). These higher regularities are essential to improve the growth condition A). The well-posedness of Cahn illiard system has been treated in many studies see, e.g., [1]). In regard to the abstract theorem of the evolution equation, we refer the reader to [16, 17]. Based on these results, we obtain the following proposition:

7 Takeshi Fukao and Taishi Motoda 7 Proposition.1. Given assumptions A1) A5) or A1), A3) with σε) = ε 1/, A5), A6), then for each ε, 1], there exists a unique weak solution of P) ε. This proposition implies that, for the well-posedness of the Cahn illiard system, the growth condition A) is not essential. It can be recovered by the strong assumption A6). The proof of this proposition is given in Section 4. Indeed, we can prove this proposition by considering the approximate problem given in Proposition 3.1. Our main theorem is now given. Theorem.1. With assumptions A1) A5), for each ε, 1], let u ε, µ ε, ξ ε ) be the weak solution of P) ε obtained in Proposition.1. Then, there exists a weak solution u, ξ) of P) and u, ξ) characterized by u ε, µ ε, ξ ε ) in the following sense: u ε u strongly in C [, T ]; V ) and weakly star in 1, T ; V ) L, T ; ), ξ ε ξ weakly in L, T ; ), µ ε µ := ξ f weakly in L, T ; V ) as ε. Moreover, the component u of the solution of P) is uniquely determined. Also, if β is single-valued, then the component ξ of the solution of P) is also unique. The second theorem relates to improving the well-posedness result of [9, 1]. Theorem.. Given assumptions A1), A3) with σε) = ε 1/, A5), A6), the same statement as in Theorem.1 holds. 3 Approximate problem and uniform estimates The proof of the main theorems exploits the characterization of the nonlinear diffusion equation through the asymptotic limits of the Cahn illiard system [7, 11]. To apply it, we consider the second-order approximation of the nonlinear term β. At this level of approximation, we obtain a uniform estimate independent of the approximation parameters. 3.1 Approximate problem for P) ε We consider an approximate problem to show the well-posedness of P) ε. For each λ, 1], we define β λ : R R by β λ r) := 1 λ r Jλ r) ) for all r R, where the resolvent operator J λ : R R is given by J λ r) := I + λβ) 1 r) for all r R. Also, we define the Moreau Yosida regularization β λ of β : R R by { } 1 β λ r) := inf s R λ r s + βs) = 1 r Jλ r) + λ β J λ r) ) for all r R.

8 8 Nonlinear diffusion equations with Robin boundary conditions Now, we consider the problem P) ε,λ for the viscous Cahn illiard like system: P) ε,λ u ε,λ µ ε,λ = a.e. in Q, t µ ε,λ = λ u ε,λ ε u ε,λ + β λ u ε,λ ) + π ε u ε,λ ) f a.e. in Q, t ν u ε,λ + κu ε,λ =, ν µ ε,λ + κµ ε,λ = a.e. on Σ, u ε,λ ) = u ε a.e. in. Define A : DA) by Au = u in with DA) = W ; the treatment of A is given in Appendix. From the well-known abstract theory of the doubly nonlinear evolution equation [8], we obtain the following well-posedness result see, also [6, 16, 17]): Proposition 3.1. For each λ, 1], there exists a unique u ε,λ 1, T ; ) L, T ; V ) L, T ; W ) such that u ε,λ satisfies the following Cauchy problem λi + F 1 )u ε,λt) + εau ε,λ t) = β λ uε,λ t) ) π ε uε,λ t) ) + ft) in, for a.a. t, T ), 3.1) u ε,λ ) = u ε in. With this level of abstractness, the Cahn illiard system with Robin boundary condition is essentially the same as in previous studies. Therefore, we omit the proof of this proposition. Now, for each λ, 1], we put µ ε,λ t) := λu ε,λs) + εau ε,λ t) + β λ uε,λ t) ) + π ε uε,λ t) ) ft) in, 3.) for a.a. t, T ). Then, we can rewrite the evolution equation 3.1) as F 1 u ε,λt) + µ ε,λ t) = in V, 3.3) for a.a. t, T ). We remark here that we do not need the projection to µ ε,λ because of the boundary condition cf. [6, 16, 17]). This is different from that for the Neumann boundary condition. 3. Uniform estimates To prove the convergence theorem, we now obtain the uniform estimates independent of ε, λ.

9 Takeshi Fukao and Taishi Motoda 9 Lemma 3.1. There exists a positive constant M 1 and two values λ, ε, 1], depending only on the data, such that t t u ε,λ s) ds + λ V u ε,λ s) ds + ε uε,λ t) V + βλ uε,λ t) ) + c 1 uε,λ L t) 1 ) 4 M 1, 3.4) t µε,λ s) ds M V 1 3.5) for all t [, T ], λ, λ] and ε, ε]. Proof We test 3.1) at time s, T ) for u ε,λ s). Then, we see that λ u ε,λs) + u ε,λs) ε u V ε,λ s), u ε,λs) ) + d β λ uε,λ s) ) dx = π ε uε,λ s) ), u ds ε,λs) ) + fs), u ε,λs) ) 3.6) for a.a. s, T ). We have now from the boundary condition of u ε,λ s) u ε,λ s), u ε,λs) ) = = κ = κ d ds ν u ε,λ s)u ε,λs)d + u ε,λ s) u ε,λs)dx u ε,λ s)u ε,λs)d + u ε,λ s) u ε,λs)dx uε,λ s) d + 1 d uε,λ s) dx ds for a.a. s, T ). Integrating 3.6)with respect to s over interval [, t], and using the above, we infer that t u ε,λs) t ds + λ u V ε,λs) ds + ε u ε,λ t) + β V λ uε,λ t) ) dx = ε u ε V + β λ u ε )dx + π ε u ε )dx π ε uε,λ t) ) dx + t u ε,λ s), fs) ds 3.7) V,V for all t [, T ], where π ε is the primitive of π ε given by π ε r) := r π ετ)dτ for all r R. Now, aided by assumption A), we have β λ r) = 1 r J λ r) + λ β λ Jλ r) ) 1 r J λ r) + c 1 J λ r) c for all r R. λ

10 1 Nonlinear diffusion equations with Robin boundary conditions ence, putting λ := min{1, 1/c 1 )}, we see that c 1 / 1/4 λ). It follows that β λ uε,λ s) ) 1 β λ uε,λ s) ) dx + 1 u ε,λ s) J λ uε,λ s) ) 4 λ + c 1 Jλ uε,λ s) ) c 1 β λ uε,λ s) ) dx + c 1 uε,λ s) 4 c 3.8) for a.a. s, T ). Next, we use the Maclaurin expansion and A3) to obtain πε r) πε ) r + 1 π ε L R)r c 3 σε)1 + r ) for all r R. Also, with assumption A3), there exists ε, 1] such that σε) c 1 8c ) for all ε, ε]; that is, we deduce that π ε uε,λ t) ) dx c 3 σε) c uε,λ t) ) dx 1 + u ε,λ t) ) 3.9) for all t [, T ]. ence, applying the Young inequality to 3.7) and collecting all of the above, we see that there exists a positive constant M 1 depending only on c 1, c, c 4,, and f L,T ;V ), independent of ε, ε], λ, λ] such that 3.4) holds. Finally, we have from 3.3) t µ ε,λ s) t ds = u V ε,λs) M V 1 for all t [, T ]. Lemma 3.. There exist positive constants M and M 3, independent of ε, ε] and λ, λ], such that for all t [, T ]. t t βλ uε,λ s) ) ds M, 3.1) ε uε,λ s) ds M ) Proof We test 3.) for times s, T ) by β λ u ε,λ s)). Then, we see that µε,λ s), β λ uε,λ s) )) = ε u ε,λ s), β λ uε,λ s) )) + βλ uε,λ s) ) + λu ε,λs) + π ε uε,λ s) ) fs), β λ uε,λ s) ))

11 Takeshi Fukao and Taishi Motoda 11 for a.a. s, T ). Calculating the first term on the right-hand side of the above equation, we infer from the boundary condition of u ε,λ s) that ε u ε,λ s), β λ uε,λ s) )) = ε β λ uε,λ s) ) u ε,λ s) dx + κε u ε,λ s)β λ uε,λ s) ) d for a.a. s, T ) because β is monotonic. ence, applying the Young inequality, we obtain βλ uε,λ s) ) µ ε,λ s) λu ε,λs) π ε uε,λ s) ) + fs), β λ uε,λ s) )) 1 βλ uε,λ s) ) µε,λ + s) + λ u ε,λ s) + πε uε,λ s) ) + ) fs) for a.a. s, T ). Integrating with respect to s over [, t] and applying Lemma 3., we see that there exists a positive constant M depending only on M 1, c 3, and f L,T ;) such that estimate 3.1) holds. Next, by comparing with 3.), we obtain 3.11) for some positive constant M 3. 4 Proof of convergence theorem We next show the existence of solution of P) ε by passing to the limit for the approximate problem P) ε,λ. 4.1 Passage to the limit λ Proof of Proposition.1. From previous estimates established in Lemmas 3.1 and 3., we see that there exists a subsequence {λ k } k N with λ k as k + and some limit functions u ε 1, T ; V ) L, T ; V ) L, T ; W ), µ ε L, T ; V ), and ξ ε L, T ; ) such that u ε,λk u ε weakly star in 1, T ; V ) L, T ; V ), 4.1) λ k u ε,λk strongly in 1, T ; ), 4.13) µ ε,λk µ ε weakly in L, T ; V ), 4.14) β λk u ε,λk ) ξ ε weakly in L, T ; ), 4.15) ε u ε,λk ε u ε weakly in L, T ; ) 4.16) as k +. From 4.1) and well-known compactness results see, e.g., []), we obtain u ε,λk u ε strongly in C [, T ]; ) 4.17) as k +. From 4.17), we deduce that u ε ) = u ε a.e. in. Moreover, from 4.17) and the Lipschitz continuity of π ε, we deduce that π ε u ε,λk ) π ε u ε ) strongly in C [, T ]; ) 4.18)

12 1 Nonlinear diffusion equations with Robin boundary conditions as k +. Also, applying 4.15), 4.17), and the monotonicity of β, we obtain ξ ε βu ε ) a.e. in Q. Finally, given the level of approximation associated with the weak formulations of 3.) 3.3), taking the limit as k +, and using 4.1) 4.18), we find that u ε, µ ε, ξ ε ) satisfies.4).5). 4. Passage to the limit ε Proof of Theorem.1. From weakly and strongly convergence, 4.1) 4.16), the same kind of uniform estimates in Lemmas 3.1 and 3. holds for u ε, µ ε, ξ ε ), that is, t u ε s) ds + ε uε t) + c 1 uε t) V V 4 M 1, t µε s) ds M V 1, t t ξε s) ds M, ε uε s) ds M 3 for all t [, T ]. ence, there exists a subsequence {ε k } k N with ε k as k + and some limits functions u 1, T ; V ) L, T ; ), µ L, T ; V ), ξ L, T ; ) such that u εk u weakly star in 1, T ; V ) L, T ; ), 4.19) ε k u εk strongly in L, T ; V ), 4.) µ εk µ weakly in L, T ; V ), 4.1) ξ εk ξ weakly in L, T ; ) 4.) as k +. From 4.19) and the well-known Ascoli Arzelà theorem see, e.g., []), we obtain u εk u strongly in C [, T ]; V ). 4.3) Moreover, by virtue of 4.), ε k u εk weakly in L, T ; ) 4.4) as k +. Aided by assumption A3), we see that πεk u εk ) π εk L R) u εk + uεk ) c3 σε k ) 1 + u εk ) a.e. in Q. Therefore, we obtain π εk u εk ) strongly in L, T ; ) 4.5) as k +. Now, from.4), we have u ε t), z V,V + F z, µ ε t) V,V = for all z V,

13 Takeshi Fukao and Taishi Motoda 13 for a.a. t, T ). Therefore, using 4.19), 4.1) and 4.3), we obtain.8) by passing to the limit in the above, that is,.1). On the other hand, in.5), using 4.1), 4.), 4.4), 4.5) and performing a comparison, we obtain µ = ξ f in L, T ; ). Moreover, by comparison, this gives us the additional regularity ξ = µ + f L, T ; V ) because µ, f L, T ; V ). We now have u C[, T ]; V ) L, T ; ); the function u is thus weakly continuous from [, T ] to, that is,.) holds. Finally, it remains to show that ξ βu) a.e. in Q. For this purpose, we show that lim sup k + T ξεk t), u εk t) ) T dt ) ξt), ut) dt. 4.6) To this aim, testing.5) for u εk t) and integrating over [, T ], we deduce from the boundary condition that T ξεk t), u εk t) ) T dt = µεk t) + ft), u εk t) ) T dt + ε k uεk t), u εk t) ) dt = T T ε k κ T πεk uεk t) ), u εk t) ) dt uεk t), µ εk t) + ft) T dt ε V,V k uεk t) dt d T From 4.1) and 4.3), we see that T lim k uεk t) dt T πεk uεk t) ), u εk t) ) dt uεk t), µ εk t) + ft) T dt V πεk uεk t) ), u,v εk t) ) dt. uεk t), µ εk t) + ft) T dt = V,V ut), µt) + ft) Moreover, from 4.19) and 4.5), we have lim k T = = T T ut), µt) + ft) ) ξt), ut) ) πεk uεk t) ), u εk t) ) dt =. dt. V,V dt Therefore, 4.6) holds. Applying the closedness theorem with respect to weakly-weakly convergence see, e.g., [5, Lemma.3]), we obtain the fact that ξ βu) a.e. in Q. Thus, the proof of existence stated in Theorem.1 is complete. Next, we show that the component u of the weak solution of P) is unique. Now, for i = 1,, let u i), ξ i) ) be weak solutions of P), respectively. Then, from.8), we have u 1 t) u t), z + V,V µ 1 t) µ t) ) zdx + κ µ1 t) µ t) ) zd = for all z V. dt

14 14 Nonlinear diffusion equations with Robin boundary conditions ere, putting z := F 1 u 1 t) u t)) in V at time t, T ), using the monotonicity of β, and µ 1 µ = ξ 1 ξ a.e. in Q, we infer that µ 1 t) µ t) ) F 1 u 1 t) u t) ) dx + κ µ1 t) µ t) ) F 1 u 1 t) u t) ) d = F F 1 u 1 t) u t) ), ξ 1 t) ξ t) V,V = u 1 t) u t), ξ 1 t) ξ t) V,V = ξ 1 t) ξ t), u 1 t) u t) ). Therefore, for all t [, T ], we deduce that 1 u1 t) u t) + V t ξ1 t) ξ t), u 1 t) u t) ) dt, which implies that the component u is unique. Thus, we obtain our result. Remark 4.1. i) Allowing the maximal monotone graph β to be multi-valued, the component ξ is therefore not uniquely determined. owever, if β is single-valued, then ξ = βu) is unique. ii) The argument for the proof of Theorem.1 is essentially the same as in [7]. Also, we can obtain the error estimates by some reinforcement. Indeed, if we add the assumption in.6) of A3) by σε) := ε 1/. Moreover, if we add the following assumption in A5), then there exists c 4 > such that Then we obtain error estimate u ε u C[,T ];V ) + u ε u V c 4 ε 1 4 for all ε, 1]. 4.7) T ξε t) ξt), u ε t) ut) ) dt C ε ) for all ε, ε], where C is a positive constant depending only on the data see, [7, Theorem 5.1]). 5 Improvement of the results As mentioned in Introduction, for all of the results from the ilbert space approach to nonlinear diffusion equation, the growth condition A) for β is a very important assumption. Nevertheless, it is too restricted from the perspective of applications. For this reason, improving the growth condition was studied in [1] using the lower semicontinuous convex extension. See also a different approach given in [4]. In this section, we consider the improved growth condition A) for β. ereafter, we assume A1), and A3) with σε) = ε 1/, A5), and A6); that is, assumption A) is avoided.

15 Takeshi Fukao and Taishi Motoda 15 Proof of Theorem.. Recall 3.3) in the following form u ε,λs) + F µ ε,λ s) = in V, for a.a. s, T ). Multiplying u ε,λ s) by the above, we see that u ε,λ s), u ε,λ s) + µ V,V ε,λ s) u ε,λ s)dx + κ µ ε,λ s)u ε,λ s)d = 5.9) for a.a. s, T ). Moreover, testing 3.) with u ε,λ s) and using boundary conditions and assumption A6), we deduce that µ ε,λ s) u ε,λ s)ds + κ µ ε,λ s)u ε,λ s)d = λ d uε,λ s) + κλ d uε,λ s) + ε uε,λ s) ds d ds + β λ uε,λ s) ) uε,λ s) dx + κ u ε,λ s)β λ uε,λ s) ) d π ε uε,λ s) ), u ε,λ s) ) + fs), u ε,λ s) ) 5.3) for a.a. s, T ). ere, we used fs), uε,λ s) ) = = = fs) u ε,λ s)dx + fs) ν u ε,λ s)d fs) u ε,λ s)dx κ fs)u ε,λ s)d fs) u ε,λ s)dx + ν fs)u ε,λ s)d = fs), u ε,λ s) ) for a.a. s, T ). ere, using A3) and the Young inequality, we infer that πε uε,λ s) ), u ε,λ s) ) π ε L R) u ε,λ s) + π ε ) ) u ε,λ s) dx c 3 ε uε,λ s) ) uε,λ s) dx ε u ε,λ s) + c 3 + u ε,λ s) ) 5.31) for a.a. s, T ). Then, combining 5.9) 5.31) and integrating the resultant with respect to s over interval [, t], we obtain 1 u ε,λ t) + λ u ε,λ t) + κλ d 1 u ε + λ u ε + κλ d u ε t + c 3 + uε,λ s) ) ds + 1 u ε,λ t) + ε t t fs) ds + 1 u ε,λ s) ds t uε,λ s) ds 5.3)

16 16 Nonlinear diffusion equations with Robin boundary conditions for all t [, T ]. Now, note that we can take λ ε, because λ tends to with fixed ε. Also, from.7) of assumption A5), we have 1 u ε + λ u ε d + κλ u ε κ)c ) Then, using 5.3), 5.33), and the Gronwall inequality, it follows that u ε,λ t) + λ u ε,λ t) + κλ u d ε,λ t) { 31 + κ)c 4 + for all t [, T ]. Moreover, ε t T fs) ds + c 3 T uε,λ s) ds M 41 + c 3T + T ) } exp { c 3 + 1)T } =: M 4 for all t [, T ]. ence, according to Lemma 3.1, using 3.7) and 3.9) without 3.8), we obtain 1 t u ε,λs) t ds + λ u V ε,λs) ds + ε u ε,λ t) + βλ uε,λ t) ) V L 1 ) 3 c 4 + c c 4) + c M 4) + 1 f L,T ;V ) for all t [, T ]. Therefore, we obtain the same kind of uniform estimates 3.4) and 3.5) in Lemma 3.1 without the growth condition A). The rest of the proof of Theorem. is the same as that of Theorem.1. Remark 5.1. As stated in Remark 4.1, we also can improve the error estimate, by assuming A1), A3) with σε) := ε 1/, A5) and 4.7). Then, 4.8) is improved by stating it in the form u ε u C[,T ];V ) + T for all ε, ε] see, [7, Theorem 6.1]). ξε t) ξt), u ε t) ut) ) dt C ε 1 6 Asymptotic limits to solutions of Neumann problem In this section, we establish the order of convergence between the Robin problem and the Neumann problem related to P). We rewrite our problem for P), hereafter denoted P) R, as follows: u κ t ξ κ = g κ, ξ κ βu κ ) in Q, P) R ν ξ κ + κξ κ = h κ on Σ, u κ ) = u κ in

17 Takeshi Fukao and Taishi Motoda 17 for κ >, where g κ : Q R, h κ : Σ R, and u κ : R are given data, and β is same maximal monotone graph as in A1). Moreover, we recall the previous result [7] for problem P) subject to the Neumann boundary condition, P) N u ξ = g, t ξ βu) in Q, ν ξ = h on Σ, u) = u in which is denoted P) N, where g : Q R, h : Σ R, and u : R are given data. The well-posedness for P) N has already discussed in [7, Theorem.3], specifically, there exists a pair u, ξ) such that u 1, T ; V ) L, T ; ), ξ L, T ; V ) with ξ βu) a.e. in Q and they satisfy u t), z V,V + ξt) zdx = gt)zdx + ht)zd for all z V, for a.a. t, T ), 6.34) u) = u a.e. in. For each κ >, we assume in addition that, A7) g, g κ L, T ; ), h, h κ L, T ; ) and u, u κ with βu ), βu κ ) L 1 ). Moreover, there exists a positive constant c 5 such that g κ g L,T ;) κc 5, h κ h L,T ; ) κc 5, u κ u V κc ) Then, we obtain the order of convergence between the Robin problem P) R and the Neumann problem P) N as κ : Theorem 6.1. Under assumptions A1), A), A7) or A1), A7) with h = h κ. Let u, ξ) be the weak solution P) N and u κ, ξ κ ) be the weak solution of P) R. Then, there exists a positive constant M, depending only the data, such that u κ u C[,T ];V ) + T ξκ s) ξs), u κ s) us) ) ds M κ 6.36) for all κ >. Moreover, if β is Lipschitz continuous, then it follows that T ξκ s) ξs) ds C βm κ for all κ >, where C β is the Lipschitz constant for β.

18 18 Nonlinear diffusion equations with Robin boundary conditions Proof We take the difference between.1) for u κ and 6.34) for u at t = s and test it setting z := F 1 u κ s) us)) at time s, T ). Then, we have 1 d uκ s) us) + ξ ds V κ s) ξs) ) F 1 u κ s) us) ) dx + κ ξ κ s)f 1 u κ s) us) ) d = gκ s) gs) ) F 1 u κ s) us) ) dx + hκ s) hs) ) F 1 u κ s) us) ) d. Now, note that ξ κ s) ξs) ) F 1 u κ s) us) ) dx + κ ξ κ s)f 1 u κ s) us) ) d = F F 1 u κ s) us) ), ξ κ s) ξs) V,V + κ ξs), F 1 u κ s) us) )) = u κ s) us), ξ κ s) ξs) ) + κ ξs), F 1 u κ s) us) )). Therefore, using the Young inequality and the trace theorem c P V, we deduce that 1 d uκ s) us) + ξ ds V κ s) ξs), u κ s) us) ) = κ ξs), F 1 u κ s) us) )) + g κ s) gs), F 1 u κ s) us) )) + h κ s) hs), F 1 u κ s) us) )) 3 κ c P ξs) + 1 F 1 u 6c κ s) us) ) + 3 gκ s) gs) P + 1 F 1 u κ s) us) ) c P hκ s) hs) + 1 F 1 u 6c κ s) us) ) P 3 κ c P ξs) + 3 gκ s) gs) V + 3 c P hκ s) hs) + 1 uκ s) us) 6.37) V for a.a. s, T ). ere, using 6.35) of A7) and the Gronwall inequality, we infer that uκ t) ut) V { } u κ u V + 3c Pκ ξ L,T ;V ) + 3 g κ g L,T ;) + 3c P h κ h L,T ; ) expt ) κ { c 5 + 3c P ξ L,T ;V ) + 3c 5 + 3c Pc 5} expt ) for all t [, T ]. Moreover, by integrating 6.37) over [, T ] with respect s, there exists a positive constant M, such that the estimates 6.36) holds. If β is Lipschitz continuous, we infer from its monotonicity that T ξκ s) ξs) T ds C uκ β s) us) ξκ s) ξs) dxds This completes the proof. T = C β ξκ s) ξs), u κ s) us) ) ds C β M κ.

19 Takeshi Fukao and Taishi Motoda 19 Acknowledgments We thank Richard aase, Ph.D, from Edanz Group for editing a draft of this manuscript. Appendix We use the same notation as in the previous sections. To characterize the operator A as in Section 3, we employ the maximal monotone theory related to the subdifferential. To do so, we define a proper lower semicontinuous convex functional ϕ : [, + ], 1 z dx + κ z d if z V, ϕz) := + otherwise. We now present the representation of subdifferential operator ϕ see e.g., [5, pp.6 65, Proposition.9]). Lemma A. The subdifferential ϕ on is characterized by ϕz) = z in for all z D ϕ) = W. Proof Let z Dϕ) = V. For all z ϕz) in. From the definition of the subdifferential ϕ, we have z, z) = z, z) d + κz, z) for all z V. Now, taking z D) as the test function, we deduce z = z in D ), that is, z = z in by comparison. ence, for all z V, we have z, z) = z, z) + ν z, z + κ z zd, specifically, by comparison, ν z = κz in 1/ ). Therefore, from the theory of elliptic regularity for Neumann problem, it follows that z ). Thus, we infer that ϕz) is singleton and is equal to z with D ϕ) = {z ) : ν z + κz = a.e. on }. Finally, we can define A by ϕ and then find the same abstract structure for the evolution equation 3.1), which has also been treated in previous works [6, 16, 17]. References [1] T. Aiki, Two-phase Stefan problems with dynamic boundary conditions, Adv. Math. Sci. Appl., 1993), 53 7.

20 Nonlinear diffusion equations with Robin boundary conditions [] T. Aiki, Multi-dimensional Stefan problems with dynamic boundary conditions, Appl. Anal., ), [3] T. Aiki, Periodic stability of solutions to some degenerate parabolic equations with dynamic boundary conditions, J. Math. Soc. Japan, ), [4] G. Akagi, Energy solutions of the Cauchy Neumann problem for porous medium equations, pp.1 1 in Discrete and Continuous Dynamical Systems, supplement 9, American Institute of Mathematical Sciences, 9. [5] V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer, London, 1. [6] P. Colli and T. Fukao, Equation and dynamic boundary condition of Cahn illiard type with singular potentials, Nonlinear Anal., 17 15), [7] P. Colli and T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn illiard systems, J. Differential Equations, 6 16), [8] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations, ), [9] A. Damlamian, Some results on the multi-phase Stefan problem, Comm. Partial Differential Equations, 1977), [1] A. Damlamian and N. Kenmochi, Evolution equations generated by subdifferentials in the dual space of 1 )), Discrete Contin. Dyn. Syst., ), [11] T. Fukao, Convergence of Cahn illiard systems to the Stefan problem with dynamic boundary conditions, Asymptot. Anal., 99 16), 1 1. [1] T. Fukao, Cahn illiard approach to some degenerate parabolic equations with dynamic boundary conditions, in: L. Bociu, J-A. Désidéri and A. abbal Eds.), System Modeling and Optimization, Springer, Switzerland, 16, pp [13] T. Fukao, S. Kurima and T. Yokota, Nonlinear diffusion equations as asymptotic limits of Cahn illiard systems on unbounded domains via Cauchy s criterion, to appear in Math. Methods Appl. Sci., 18), 1 1. [14] T. Fukao and T. Motoda, Abstract approach of degenerate parabolic equations with dynamic boundary conditions, to appear in Adv. Math. Sci. Appl., 7 18), [15] N. Kenmochi, Neumann problems for a class of nonlinear degenerate parabolic equations, Differential Integral Equations, 199), [16] N. Kenmochi, M. Niezgódka and I. Paw low, Subdifferential operator approach to the Cahn illiard equation with constraint, J. Differential Equations, ),

21 Takeshi Fukao and Taishi Motoda 1 [17] M. Kubo, The Cahn illiard equation with time-dependent constraint, Nonlinear Anal., 75 1), [18] M. Kubo and Q. Lu, Nonlinear degenerate parabolic equations with Neumann boundary condition, J. Math. Anal. Appl., 37 5), [19] S. Kurima and T. Yokota, Monotonicity methods for nonlinear diffusion equations and their approximations with error estimates, J. Differential Equations, 63 17), 4 5. [] S. Kurima and T. Yokota, A direct approach to quasilinear parabolic equations on unbounded domains by Brézis s theory for subdifferential operators, Adv. Math. Sci. Appl., 6 17), 1 4. [1] A. Miranville, The Cahn illiard equation and some of its variants, AIMS Mathematics., 17), [] J. Simon, Compact sets in the spaces L p, T ; B), Ann. Mat. Pura. Appl. 4), ),