A free boundary problem describing migration into rubbers -- quest of the large time behavior
AA free boundary problem describing migration intorubbers – quest of the large time behavior
Kota Kumazaki
Faculty of Education, Nagasaki University,1-14 Bunkyo-cho, Nagasaki-city, Nagasaki, 851-8521, [email protected]
Toyohiko Aiki
Department of Mathematics Faculty of Sciences, Japan Women’s University,2-8-1, Mejirodai, Bunkyo-ku, Tokyo, 112-8681, [email protected]
Adrian Muntean
Department of Mathematics and Computer Science, Karlstad University,Universitetsgatan 2, 651 88 Karlstad, [email protected]
Abstract
In many industrial applications, rubber-based materials are routinely used in conjunctionwith various penetrants or diluents in gaseous or liquid form. It is of interest to estimate theo-retically the penetration depth as well as the amount of diffusants stored inside the material. Inthis framework, we prove the global solvability and explore the large time-behavior of solutionsto a one-phase free boundary problem with nonlinear kinetic condition that is able to describethe migration of diffusants into rubber. The key idea in the proof of the large time behavior isto benefit of a contradiction argument, since it is difficult to obtain uniform estimates for thegrowth rate of the free boundary due to the use of a Robin boundary condition posed at the fixedboundary.
In many industrial applications, the behavior of rubber-based materials is difficult to predict theoret-ically. This intriguing fact is especially due to their internal structure which allows for unexpectedlocal changes (deformations, concentration localization, network entaglement, etc.) typically fa-cilitated by the absorption and migration of diffusants into the material; this is from where ourmotivation stems. There is a variety of possible modeling approaches for such scenarios. Motivatedby our recent work [17], where solutions to our free boundary model did recover experimental data,1 a r X i v : . [ m a t h . A P ] F e b Kota Kumazaki, Toyohiko Aiki and Adrian Muntean we choose to follow a macroscopic modeling approach with kinetically-driven interfaces capturingthe penetration of diffusants into the material. We refer the reader, for instance, to [8, 9, 11, 7] forclosely related work especially what concerns the mathematics of Case II diffusion as it arises forsome classes of polymers, but not directly applicable to the rubber case.In this paper we consider the mathematical analysis of the following free boundary problemwhich was discussed in [17] in connection with the absorption, penetration and diffusion-inducedswelling in dense and foamed rubbers. Let [0 , s ( t )] be a region occupied by a solvent (e.g. water, tea)occupying the one-dimensional pore [0 , ∞ ) , where t is the time variable, s = s ( t ) is the positionof the moving interface, while u = u ( t, z ) is the content of the diffusant situated at the position z ∈ [0 , s ( t )] . The function u ( t, z ) acts in the non-cylindrical region Q s ( T ) given by Q s ( T ) := { ( t, z ) | < t < T, < z < s ( t ) } . Our free boundary problem ( P )( u , s , b ) reads: Find the pair ( u, s ) satisfying u t − u zz = 0 for ( t, z ) ∈ Q s ( T ) , (1.1) − u z ( t,
0) = β ( b ( t ) − γu ( t, for t ∈ (0 , T ) , (1.2) − u z ( t, s ( t )) = u ( t, s ( t )) s t ( t ) for t ∈ (0 , T ) , (1.3) s t ( t ) = a σ ( u ( t, s ( t ))) for t ∈ (0 , T ) , (1.4) s (0) = s , u (0 , z ) = u ( z ) for z ∈ [0 , s ] , (1.5)where β , γ and a are given positive constants, b is a given threshold function defined on [0 , T ] ,while s and u are the corresponding initial data. In (1.4), σ is a function on R given by σ ( r ) = (cid:40) r if r ≥ , if r < . In [18], A. Visintin refers to this type of problems as free or moving boundary problems with kinetic boundary condition. The reason for calling this way is linked to the fact that relation (1.4)is an explicit description of the speed of the free boundary. Note also that, in Refs. [15, 16], theauthors have considered the mathematical analysis of a similar problem to ( P )( u , s , b ) related towater-induced swelling in porous materials, viz. u t − u zz = 0 for ( t, z ) ∈ (0 , T ) × ( a, s ( t )) , (1.6) − u z ( t, a ) = ˆ β ( h ( t ) − γu ( t, a )) for t ∈ (0 , T ) , (1.7) − u z ( t, s ( t )) = u ( t, s ( t )) s t ( t ) for t ∈ (0 , T ) , (1.8) s t ( t ) = a ( u ( t, s ( t )) − ϕ ( s ( t ))) for t ∈ (0 , T ) , (1.9) s (0) = ˆ s , u (0 , z ) = ˆ u ( z ) for z ∈ [ a, ˆ s ] . (1.10)In this context, a is a positive constant, h is a given non-negative function on [0 , T ] , while ˆ s and ˆ u are the initial data such that ˆ s > a . Also, ˆ β and ϕ are continuous functions on R such that ˆ β ( r ) > free boundary problem describing migration into rubbers – quest of the large time behavior ϕ ( r ) > for r > , and ˆ β ( r ) = ϕ ( r ) = 0 for r ≤ . Denote the above problem { (1 . − (1 . } by (ˆ P )(ˆ u , ˆ s , h ) . In [15] in was assumed that ϕ is conveniently small and ˆ u ∈ H ( a, ˆ s ) such that ϕ ( a ) ≤ ˆ u ≤ h ∗ /γ on [ a, ˆ s ] . Such conditions ensure the existence of a locally-in-time solution ( u, s ) to (ˆ P )(ˆ u , ˆ s , h ) on [0 , T ] such that ϕ ( a ) ≤ u ≤ h ∗ /γ on Q as ( T ) for some < T ≤ T ,where h ∗ is a upper bound of h . In [16], relying on the same assumptions as in [15], the authors haveconstructed a globally-in-time solution ( u, s ) to (ˆ P )(ˆ u , ˆ s , h ) on [0 , T ] such that ϕ ( a ) ≤ u ≤ h ∗ /γ on Q as ( T ) . What concerns the large time behavior of solutions to (ˆ P )(ˆ u , ˆ s , h ) , one reports in [16]that the following situation holds:if lim t →∞ (cid:90) t ˆ β ( h ( τ ) − γu ( τ, a )) dτ = ∞ , then lim t →∞ s ( t ) = ∞ . One of our concrete aims here is to construct a global-in-time solution of ( P )( u , s , b ) . Asanticipated, the key to the proof is to establish the strictly positivity for the free boundary. To thisend, we consider the free boundary condition (1.4), which should be seen as ϕ ≡ in (1.9) in themodels proposed in [15, 16]. This is a simplification of the modeling setting which is convenient formathematical analysis purposes. Furthermore, we adopt the positive part in (1.4), and hence, we caneasily show that the free boundary s ( t ) is indeed strictly positive and the expected global existenceis now reachable. We will investigate elsewhere to which extent such structural restrictions can berelaxed.Moreover, we establish that our free boundary grows up, namely, it is unbounded. In order toobtain a control on the growth of the free boundary, the mass conservation law (respectively, themomentum balance law) are effective ingredients in case the boundary condition at the fixed bound-ary of Neumann (respectively, Dirichlet) type, see for instance [5]. In the present setting, we imposea Robin boundary condition at the fixed boundary and the usual approach does not work well.Hence, the rationale beyond showing that s ( t ) → ∞ as t → ∞ is as follows: If the free boundaryis bounded, then we can obtain some uniform-in-time estimates for the target solution, and conse-quently, this solution u ( t ) converges towards the stationary solution of our problem. It is worthwhileto note here that the stationary problem still contains a free boundary condition. However, as a con-sequence of our uniform estimates, the solution never satisfies the stationary free boundary condi-tion. Thus we can prove the large time behavior by detecting a contradiction. The idea of applyinga contradiction argument concerned with a stationary solution was already applied in [1]. The questfor growth (convergence) rates of this kind of kinetically-driven free boundaries was completed inthe series of papers [2, 3, 4]. For the problem at hand, proving quantitative estimates on the growthrate of the free boundary is currently an open problem. In this paper, we use the following notations. We denote by | · | X the norm for a Banach space X .The norm and the inner product of a Hilbert space H are denoted by | · | H and ( · , · ) H , respectively. Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
Particularly, for Ω ⊂ R , we use the notation of the usual Hilbert spaces L (Ω) , H (Ω) and H (Ω) .Throughout this paper, we assume the following parameters and functions:(A1) a , γ , β and T are positive constants.(A2) b ∈ W , (0 , T ) with b ∗ ≤ b ≤ b ∗ on (0 , T ) , where b ∗ and b ∗ are positive constants.(A3) s > and u ∈ H (0 , s ) such that ≤ u ≤ b ∗ /γ on [0 , s ] .Next, we define our concept of solution to (P) ( u , s , b ) on [0 , T ] in the following way: Definition 2.1.
For
T > , let s be a function on [0 , T ] and u be a function on Q s ( T ) . We call thepair ( s, u ) a solution to (P) ( u , s , b ) on [0 , T ] if the following conditions (S1)-(S6) hold:(S1) s ∈ W , ∞ (0 , T ) , < s on [0 , T ] , u ∈ L ∞ ( Q s ( T )) , u t , u zz ∈ L ( Q s ( T )) and t ∈ [0 , T ] →| u z ( t, · ) | L (0 ,s ( t )) is bounded;(S2) u t − u zz = 0 on Q s ( T ) ;(S3) − u z ( t,
0) = β ( b ( t ) − γu ( t, for a.e. t ∈ [0 , T ] ;(S4) − u z ( t, s ( t )) = u ( t, s ( t )) s t ( t ) for a.e. t ∈ [0 , T ] ;(S5) s t ( t ) = a σ ( u ( t, s ( t ))) for a.e. t ∈ [0 , T ] ;(S6) s (0) = s and u (0 , z ) = u ( z ) for z ∈ [0 , s ] .The first result of this paper is concerned with the existence and uniqueness of a locally-in-timesolution in the sense of Definition 2.1 to the problem (P) ( u , s , b ) . Theorem 2.2.
Let
T > . If (A1)-(A3) hold, then there exists T ∗ ∈ (0 , T ] such that (P) ( u , s , b ) has a unique solution ( s, u ) on [0 , T ∗ ] satisfying ≤ u ≤ b ∗ /γ on Q s ( T ∗ ) .To prove Theorem 2.2, we transform (P) ( u , s , b ) , initially posed in a non-cylindrical domain,to a cylindrical domain. Let T > . For given s ∈ W , (0 , T ) with s ( t ) > on [0 , T ] , we introducethe following new function obtained by the change of variables and fix the moving domain: ˜ u ( t, y ) = u ( t, ys ( t )) for ( t, y ) ∈ Q ( T ) := (0 , T ) × (0 , (2.1)By using the function ˜ u , (P) ( u , s , b ) becomes the following problem (PC) (˜ u , s , b ) on the cylin-drical domain Q ( T ) : ˜ u t ( t, y ) − s ( t ) ˜ u yy ( t, y ) = ys t ( t ) s ( t ) ˜ u y ( t, y ) for ( t, y ) ∈ Q ( T ) , (2.2) − s ( t ) ˜ u y ( t,
0) = β ( b ( t ) − γ ˜ u ( t, for t ∈ (0 , T ) , (2.3) − s ( t ) ˜ u y ( t,
1) = ˜ u ( t, s t ( t ) for t ∈ (0 , T ) , (2.4) s t ( t ) = a σ (˜ u ( t, for t ∈ (0 , T ) , (2.5) s (0) = s , (2.6) ˜ u (0 , y ) = u ( ys (0))(:= ˜ u ( y )) for y ∈ [0 , . (2.7) free boundary problem describing migration into rubbers – quest of the large time behavior Definition 2.3.
For
T > , let s be a function on [0 , T ] and ˜ u be a function on Q ( T ) , respectively.We call that a pair ( s, ˜ u ) is a solution of ( P )(˜ u , s , b ) on [0 , T ] if the conditions (S’1)-(S’2) hold:(S’1) s ∈ W , ∞ (0 , T ) , s > on [0 , T ] , ˜ u ∈ W , ( Q ( T )) ∩ L ∞ (0 , T ; H (0 , ∩ L (0 , T ; H (0 , .(S’2) (2.2)–(2.7) hold.Here, we introduce the following function space: For T > , we put V ( T ) = L ∞ (0 , T ; L (0 , ∩ L (0 , T ; H (0 , and | z | V ( T ) = | z | L ∞ (0 ,T ; L (0 , + | z y | L (0 ,T ; L (0 , for z ∈ V ( T ) . Note that V ( T ) is a Banach space with the norm | · | V ( T ) .Now, we state the existence and uniqueness of a locally-in-time solution of ( PC )(˜ u , s , b ) . Theorem 2.4.
Let
T > . If (A1)-(A3) hold, then there exists T ∗ ∈ (0 , T ] such that ( PC )(˜ u , s , b ) has a unique solution ( s, ˜ u ) on [0 , T ∗ ] .By Theorem 2.4, we see that for a solution ( s, ˜ u ) of ( PC )(˜ u , s , b ) on [0 , T ∗ ] , a pair of thefunction ( s, u ) with the variable u ( t, z ) := ˜ u (cid:18) t, zs ( t ) (cid:19) for z ∈ [0 , s ( t )] (2.8)is a solution of ( P )( u , s , b ) on [0 , T ∗ ] . Moreover, by proving that ( s, u ) satisfies ≤ u ≤ b ∗ /γ on Q s ( T ∗ ) , the pair ( s, u ) is a desired solution of ( P )( u , s , b ) on [0 , T ∗ ] which leads to Theorem 2.2.The second result of this paper is the existence and uniqueness of a globally-in-time solution inthe sense of Definition 2.1 to the problem (P) ( u , s , b ) . Theorem 2.5.
Let
T > . If (A1)-(A3) hold, then ( P )( u , s , b ) has a unique solution ( s, u ) on [0 , T ] satisfying ≤ u ≤ b ∗ /γ on Q s ( T ) .Throughout Sections 3 and 4, we show Theorem 2.2 by proving Theorem 2.4 and the bounded-ness of a solution of ( P )( u , s , b ) . In Section 5, we give a proof of Theorem 2.5. In the last section,we discuss the large time behavior of a solution of ( P )( u , s , b ) as t → ∞ . In fact, we obtain theresult that s → ∞ as t → ∞ . The precise statement is stated as Theorem 6.2. In this section, we prove Theorem 2.4 on the existence and uniqueness of a locally-in-time solutionof ( PC )(˜ u , s, b ) . To do so, we introduce the following auxiliary problem ( AP )(˜ u , s, b ) : For T > , s > and given s ∈ W , (0 , T ) with s (0) = s and s ≥ s on [0 , T ] , ˜ u t ( t, y ) − s ( t ) ˜ u yy ( t, y ) = ys t ( t ) s ( t ) ˜ u y ( t, y ) for ( t, y ) ∈ Q ( T ) , (3.1) − s ( t ) ˜ u y ( t,
0) = β ( b ( t ) − γ ˜ u ( t, for t ∈ (0 , T ) , (3.2) − s ( t ) ˜ u y ( t,
1) = a ˜ u ( t, σ (˜ u ( t, for t ∈ (0 , T ) , (3.3) ˜ u (0 , y ) = ˜ u ( y ) for y ∈ [0 , , (3.4) Kota Kumazaki, Toyohiko Aiki and Adrian Muntean where σ is the same function as in (1.4).First of all, to solve ( AP )(˜ u , s, b ) , for given s ∈ W , ∞ (0 , T ) with s (0) = s and s ≥ s on [0 , T ] and f ∈ L (0 , T ; H (0 , , we consider the problem ( AP u , s, f, b ) : ˜ u t ( t, y ) − s ( t ) ˜ u yy ( t, y ) = ys t ( t ) s ( t ) f y ( t, y ) for ( t, y ) ∈ Q ( T ) , − s ( t ) ˜ u y ( t,
0) = β ( b ( t ) − γ ˜ u ( t, for t ∈ (0 , T ) , − s ( t ) ˜ u y ( t,
1) = a ˜ u ( t, σ (˜ u ( t, for t ∈ (0 , T ) , ˜ u (0 , y ) = ˜ u ( y ) for y ∈ [0 , . Now, we define a family { ψ t } t ∈ [0 ,T ] of time-dependent functionals ψ t : L (0 , → R ∪ { + ∞} for t ∈ [0 , T ] as follows: ψ t ( u ) := s ( t ) (cid:90) | u y ( y ) | dy + 1 s ( t ) (cid:90) u (1)0 a ξσ ( ξ ) dξ − s ( t ) (cid:90) u (0)0 β ( b ( t ) − γξ ) dξ if u ∈ D ( ψ t ) , + ∞ otherwise , where D ( ψ t ) = { z ∈ H (0 , | z ≥ on [0 , } for t ∈ [0 , T ] . Here, we show the property of ψ t . Lemma 3.1.
Let s ∈ W , (0 , T ) with s (0) = s and s ≥ s on [0 , T ] and assume (A1)-(A3). Thenthe following statements hold:(1) There exists positive constants C and C such that the following inequalities hold: ( i ) | u ( y ) | ≤ C ψ t ( u ) + C for u ∈ D ( ψ t ) , y = 0 , and t ∈ [0 , T ] , ( ii ) 12 s ( t ) | u y | L (0 , ≤ C ψ t ( u ) + C for u ∈ D ( ψ t ) and t ∈ [0 , T ] . (2) For t ∈ [0 , T ] , the functional ψ t is proper, lower semi-continuous, and convex on L (0 , . Proof.
First, we note that for t ∈ [0 , T ] if u ∈ D ( ψ t ) , then u (0) is non-negative. Let t ∈ [0 , T ] and u ∈ D ( ψ t ) . Then, it holds − s ( t ) (cid:90) u (0)0 β ( b ( t ) − γξ ) dξ = βs ( t ) (cid:104) γ u (0) − b ( t ) u (0) (cid:105) ≥ βγ l u (0) − βb ∗ s u (0) ≥ βγ l u (0) − βlγ (cid:18) b ∗ s (cid:19) . (3.5)Since the second term of the right-hand side of ψ t is positive, by (3.5), we have that ψ t ( u ) ≥ s ( t ) (cid:90) | u y ( y ) | dy + βγ l u (0) − βlγ (cid:18) b ∗ s (cid:19) . (3.6) free boundary problem describing migration into rubbers – quest of the large time behavior | u (1) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) u y ( y ) dy + u (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)(cid:90) | u y ( y ) | dy + | u (0) | (cid:19) ≤ (cid:18) l s ( t ) (cid:90) | u y ( y ) | dy + | u (0) | (cid:19) . (3.7)Therefore, by (3.6) and (3.7) we see that the statement (1) of Lemma 3.1 holds.Next, we prove statement (2). For t ∈ [0 , T ] and r ∈ R , put g ( s ( t ) , r ) = 1 s ( t ) (cid:90) r a ξσ ( ξ ) dξ,g ( s ( t ) , b ( t ) , r ) = − s ( t ) (cid:90) r β ( b ( t ) − γξ ) dξ. Then, by < s ( t ) we see that ∂ ∂r g ( s ( t ) , r ) = 2 a s ( t ) r > for r > ,∂ ∂r g ( s ( t ) , b ( t ) , r ) = βγs ( t ) > for r ∈ R . This means that ψ t is convex on L (0 , . Also, by using Lemma 3.1 and Sobolev’s embedding H (0 , (cid:44) → C ([0 , in one dimensional case, it is easy to prove that the level set of ψ t is closed in L (0 , which leads to the lower semi-continuity of ψ t . Thus, we see that statement (2) holds.By Lemma 3.1 we obtain the existence of a solution to ( AP u , s, f, b ) . Lemma 3.2.
Let
T > and s > . If (A1)-(A3) hold, then, for given s ∈ W , (0 , T ) with s (0) = s and s ≥ s on [0 , T ] and f ∈ L (0 , T ; H (0 , , the problem ( AP u , s, f, b ) admits aunique solution ˜ u on [0 , T ] such that ˜ u ∈ W , ( Q ( T )) ∩ L ∞ (0 , T ; H (0 , with ˜ u ≥ on Q ( T ) .Moreover, the function t → ψ t (˜ u ( t )) is absolutely continuous on [0 , T ] . Proof.
By Lemma 3.1, for t ∈ [0 , T ] ψ t is a proper lower semi-continuous convex function on L (0 , . From the definition of the subdifferential of ψ t , for t ∈ [0 , T ] , z ∗ ∈ ∂ψ t ( u ) is characterizedby u , z ∗ ∈ L (0 , , z ∗ = − s ( t ) u yy on (0 , , − s ( t ) u y (0) = β ( b ( t ) − γu (0)) , − s ( t ) u y (1) = a u (1) σ ( u (1)) . Namely, ∂ψ t is single-valued. Also, we see that there exists a positive constant C such that for each t , t ∈ [0 , T ] with t ≤ t , and for any u ∈ D ( ψ t ) , there exists ¯ u ∈ D ( ψ t ) such that | ψ t (¯ u ) − ψ t ( u ) | ≤ C ( | s ( t ) − s ( t ) | + | b ( t ) − b ( t ) | )(1 + | ψ t ( u ) | ) . (3.8) Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
Indeed, by taking ¯ u := u and using (i) and (ii) of Lemma 3.1, we can find C > such that (3.8)holds. Now, we consider the following Cauchy problem (CP): (cid:40) ˜ u t + ∂ψ t (˜ u ( t )) = ys t ( t ) s ( t ) f y ( t ) in L (0 , , ˜ u (0 , y ) = ˜ u ( y ) for y ∈ [0 , . Since ys t s f y ∈ L (0 , T ; L (0 , , the general theory of evolution equations governed by time de-pendent subdifferentials (cf. [14]) guarantees that (CP) has a non-negative solution ˜ u on [0 , T ] suchthat ˜ u ∈ W , ( Q ( T )) , ψ t (˜ u ( t )) ∈ L ∞ (0 , T ) and t → ψ t (˜ u ( t )) is absolutely continuous on [0 , T ] .This implies that ˜ u is a unique solution of ( AP u , s, f, b ) on [0 , T ] . Lemma 3.3.
Let
T > , s > and s ∈ W , ∞ (0 , T ) with s (0) = s and s ≥ s on [0 , T ] . If(A1)-(A3) hold, then, ( AP )(˜ u , s, b ) has a unique solution ˜ u on [0 , T ] such that ˜ u ∈ W , ( Q ( T )) ∩ L ∞ (0 , T ; H (0 , . Proof.
First, we define a solution operator Γ T ( f ) = ˜ u , where ˜ u is the unique solution of ( AP u , s, f, b ) for given f ∈ V ( T ) . Now, for i = 1 , we put Γ T ( f i ) = ˜ u i and f = f − f and ˜ u = ˜ u − ˜ u . Then,we have that ddt | ˜ u ( t ) | L (0 , − (cid:90) s ( t ) ˜ u yy ( t )˜ u ( t ) dy = (cid:90) ys t ( t ) s ( t ) f y ( t )˜ u ( t ) dy. (3.9)Using the boundary condition, it holds that − (cid:90) s ( t ) ˜ u yy ( t )˜ u ( t ) dy = 1 s ( t ) (cid:18) − ˜ u y ( t, u ( t,
1) + ˜ u y ( t, u ( t,
0) + (cid:90) | ˜ u y ( t ) | dy (cid:19) = a s ( t ) (cid:18) ˜ u ( t, σ (˜ u ( t, − ˜ u ( t, σ (˜ u ( t, (cid:19) ˜ u ( t, − s ( t ) (cid:18) β ( b ( t ) − γ ˜ u ( t, − β ( b ( t ) − γ ˜ u ( t, (cid:19) ˜ u ( t,
0) + 1 s ( t ) (cid:90) | ˜ u y ( t ) | dy. (3.10)Since the function rσ ( r ) is monotone for r ∈ R , the first term of the right-hand side of (3 . isnon-negative. The second term of the right-hand side of (3.10) is also non-negative, hence we seethat − (cid:90) s ( t ) ˜ u yy ( t )˜ u ( t ) dy ≥ s ( t ) (cid:90) | ˜ u y ( t ) | dy. (3.11)Accordingly, by (3.9)-(3.11), we have that ddt | ˜ u ( t ) | L (0 , + 1 s ( t ) (cid:90) | ˜ u y ( t ) | dy ≤ (cid:90) ys t ( t ) s ( t ) f y ( t )˜ u ( t ) dy. (3.12) free boundary problem describing migration into rubbers – quest of the large time behavior (cid:90) ys t ( t ) s ( t ) f y ( t )˜ u ( t ) dy ≤ | s t | L ∞ (0 ,T ) s | ˜ u ( t ) | L (0 , | f y ( t ) | L (0 , . (3.13)Let T ∈ (0 , T ] . Then, by putting l = max ≤ t ≤ T | s ( t ) | and integrating (3.12) with (3.13) over [0 , t ] for any t ∈ [0 , T ] we obtain that | ˜ u ( t ) | L (0 , + 12 l (cid:90) t (cid:90) | ˜ u y ( τ ) | dydτ ≤ | s t | L ∞ (0 ,T ) s | ˜ u | L ∞ (0 ,T ; L (0 , T / (cid:18)(cid:90) T | f y ( τ ) | L (0 , dτ (cid:19) / ≤ | s t | L ∞ (0 ,T ) s | ˜ u | V ( T ) T / | f | V ( T ) (3.14)Therefore, by putting δ = min { / , / l } in (3.14) we have that δ | ˜ u | V ( T ) ≤ | s t | L ∞ (0 ,T ) s T / | f | V ( T ) for T ∈ (0 , T ] . From this result, we infer that for some T ≤ T such that Γ T is a contraction mapping in V ( T ) .Therefore, by Banach’s fixed point theorem there exists ˜ u ∈ V ( T ) such that Γ T (˜ u ) = ˜ u whichimplies ˜ u is a solution of ( AP )(˜ u , s, b ) on [0 , T ] . Since T is independent of the choice of initialdata of ˜ u , by repeating the argument of the local existence result, we can extend the solution ˜ u beyond T . Thus, we prove that Lemma 3.3 holds.Next, for given s ∈ W , (0 , T ) with s (0) = s and s ≥ s on [0 , T ] , we construct a solution to ( AP )(˜ u , s, b ) . Lemma 3.4.
Let
T > and s > . If (A1)-(A3) hold, then, for given s ∈ W , (0 , T ) with s (0) = s and s ≥ s on [0 , T ] , ( AP )(˜ u , s, b ) has a unique solution ˜ u on [0 , T ] . Proof.
For given s ∈ W , (0 , T ) with s (0) = s and s ≥ s on [0 , T ] , we choose a sequence { s n } ⊂ W , ∞ (0 , T ) and l > satisfying s ≤ s n ≤ l on [0 , T ] for each n ∈ N , s n → s in W , (0 , T ) as n → ∞ . By Lemma 3.3 we can take a sequence { ˜ u n } of solutions to ( AP )(˜ u , s n , b ) on [0 , T ] .Then, we see that t → ψ t (˜ u n ( t )) is absolutely continuous on [0 , T ] so that t → s n ( t ) | ˜ u ny ( t ) | L (0 , iscontinuous on [0 , T ] . First, it holds that ddt | ˜ u n ( t ) | L (0 , − (cid:90) s n ( t ) ˜ u nyy ( t )˜ u n ( t ) dy = (cid:90) ys nt ( t ) s n ( t ) ˜ u ny ( t )˜ u n ( t ) dy. For the second term in the left-hand side, we have that − (cid:90) s n ( t ) ˜ u nyy ( t )˜ u n ( t ) dy = a s n ( t ) ˜ u n ( t, σ (˜ u n ( t, u n ( t, − s n ( t ) β ( b ( t ) − γ ˜ u n ( t, u n ( t,
0) + 1 s n ( t ) (cid:90) | ˜ u ny ( t ) | dy ≥ − s n ( t ) β ( b ( t ) − γ ˜ u n ( t, u n ( t,
0) + 1 s n ( t ) (cid:90) | ˜ u ny ( t ) | dy. Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
Hence, we obtain that ddt | ˜ u n ( t ) | L (0 , + 1 s n ( t ) (cid:90) | ˜ u ny ( t ) | dy ≤ (cid:90) ys nt ( t ) s n ( t ) ˜ u ny ( t )˜ u n ( t ) dy + 1 s n ( t ) β ( b ( t ) − γ ˜ u n ( t, u n ( t, for t ∈ [0 , T ] . (3.15)Using Young’s inequality, we have that (cid:90) ys nt ( t ) s n ( t ) ˜ u ny ( t )˜ u n ( t ) dy ≤ s n ( t ) (cid:90) | ˜ u ny ( t ) | dy + | s nt ( t ) | (cid:90) | ˜ u n ( t ) | dy. (3.16)Here, by Sobolev’s embedding theorem in one dimension, we note that it holds that | v ( t, y ) | ≤ C e | v ( t ) | H (0 , | v ( t ) | L (0 , for v ∈ H (0 , and y ∈ [0 , , (3.17)where C e is a positive constant defined from Sobolev’s embedding theorem. By (3.17) and s ≤ s n on [0 , T ] we get s n ( t ) β ( b ( t ) − γ ˜ u n ( t, u n ( t, ≤ βb ∗ s n ( t ) | ˜ u n ( t, |≤ βb ∗ C e s n ( t ) (cid:18) | ˜ u ny ( t ) | L (0 , | ˜ u n ( t ) | L (0 , + | ˜ u n ( t ) | L (0 , (cid:19) + βb ∗ s n ( t ) ≤ s n ( t ) | ˜ u ny ( t ) | L (0 , + (cid:18) ( βb ∗ C e ) + βb ∗ C e s (cid:19) | ˜ u n ( t ) | L (0 , + βb ∗ s . (3.18)As a result, we see from (3.15)-(3.18) that ddt | ˜ u n ( t ) | L (0 , + 14 s n ( t ) (cid:90) | ˜ u ny ( τ ) | dy ≤ (cid:18) | s nt ( t ) | βb ∗ C e ) + βb ∗ C e s (cid:19) | ˜ u n ( t ) | L (0 , + βb ∗ s for t ∈ [0 , T ] . Now, we denote F n ( t ) the coefficient of | ˜ u n | L (0 , in the right-hand side. Then, by s n ≤ l on [0 , T ] ,the boundedness of { F n } in L (0 , T ) and Gronwall’s inequality we obtain that | ˜ u n ( t ) | L (0 , + 14 l (cid:90) t | ˜ u ny ( t ) | L (0 , dτ ≤ (cid:18) | ˜ u | L (0 , + (cid:18) βb ∗ s (cid:19) T (cid:19) e C for t ∈ [0 , T ] . (3.19)Next, we put ˜ u n ( t ) = u for t < . For each n ∈ N and h > , it holds (cid:90) ˜ u nt ( t ) ˜ u n ( t ) − ˜ u n ( t − h ) h dy − (cid:90) s n ( t ) ˜ u nyy ( t ) ˜ u n ( t ) − ˜ u n ( t − h ) h dy = (cid:90) ys nt ( t ) s n ( t ) ˜ u ny ( t ) ˜ u n ( t ) − ˜ u n ( t − h ) h dy. (3.20) free boundary problem describing migration into rubbers – quest of the large time behavior − (cid:90) s ( t ) ˜ u nyy ( t ) ˜ u n ( t ) − ˜ u n ( t − h ) h dy = − ˜ u ny ( t, s n ( t ) ˜ u n ( t, − ˜ u n ( t − h, h + ˜ u ny ( t, s n ( t ) ˜ u n ( t, − ˜ u n ( t − h, h + (cid:90) ˜ u ny ( t ) s n ( t ) ˜ u ny ( t ) − ˜ u ny ( t − h ) h dy. We denote I , I and I the three terms in the last identity and estimate three terms separately. For I , using the same notation g and g in the proof of Lemma 3.1, it follows that I ≥ h s n ( t ) (cid:32)(cid:90) ˜ u n ( t, a ξσ ( ξ ) dξ − (cid:90) ˜ u n ( t − h, a ξσ ( ξ ) dξ (cid:33) = g ( s n ( t ) , ˜ u n ( t, − g ( s n ( t − h ) , ˜ u n ( t − h, h + 1 h (cid:18) s n ( t − h ) − s n ( t ) (cid:19) (cid:90) ˜ u n ( t − h, a ξσ ( ξ ) dξ. Next, for I and I we have that I ≥ h s n ( t ) (cid:32) − (cid:90) ˜ u n ( t, β ( b ( t ) − γξ ) dξ + (cid:90) ˜ u n ( t − h, β ( b ( t ) − γξ ) dξ (cid:33) = g ( s n ( t ) , b ( t ) , ˜ u n ( t, − g ( s n ( t − h ) , b ( t − h ) , ˜ u n ( t − h, h + 1 h (cid:18) − s n ( t − h ) + 1 s n ( t ) (cid:19) (cid:90) ˜ u n ( t − h, β ( b ( t − h ) − γξ ) dξ − h s n ( t ) (cid:90) ˜ u n ( t − h, (cid:18) β ( b ( t − h ) − γξ ) − β ( b ( t ) − γξ ) (cid:19) dξ, and I ≥ h s n ( t ) (cid:18)(cid:90) | ˜ u ny ( t ) | dy − (cid:90) | ˜ u ny ( t − h ) | dy (cid:19) = 1 h (cid:18) s n ( t ) (cid:90) | ˜ u ny ( t ) | dy − s n ( t − h ) (cid:90) | ˜ u ny ( t − h ) | dy (cid:19) + 1 h (cid:18) s n ( t − h ) − s n ( t ) (cid:19) (cid:90) | ˜ u ny ( t − h ) | dy. Combining the above three estimates and using the fact that t → /s n ( t ) | ˜ u ny ( t ) | is continuous on [0 , T ] , we obtain lim inf h → ( I + I + I ) ≥ ddt ψ t (˜ u n ( t )) + s nt ( t ) s n ( t ) (cid:90) ˜ u n ( t, a ξσ ( ξ ) dξ + s nt ( t ) s n ( t ) (cid:90) ˜ u n ( t, β ( b ( t ) − γξ ) dξ + 1 s n ( t ) (cid:90) ˜ u n ( t, βb t ( t ) dξ + s nt ( t ) s n ( t ) (cid:90) | ˜ u ny ( t ) | dy. Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
Applying this result to (3.20) and letting h → , we observe that | ˜ u nt ( t ) | L (0 , + ddt ψ t (˜ u n ( t )) ≤ (cid:90) ys nt ( t ) s n ( t ) ˜ u ny ( t )˜ u nt ( t ) dy + | s nt ( t ) | s n ( t ) (cid:90) ˜ u n ( t, a ξσ ( ξ ) dξ + | s nt ( t ) | s n ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ˜ u n ( t, β ( b ( t ) − γξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) + 1 s n ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ˜ u n ( t, βb t ( t ) dξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | s nt ( t ) | s n ( t ) (cid:90) | ˜ u ny ( t ) | dy. (3.21)Denote J i (1 ≤ i ≤ each terms in the right-hand side of (3.21). Using Lemma 3.1 and s n ≥ s on [0 , T ] , we estimate each terms J i except for i = 2 as follows: J ≤ | ˜ u nt ( t ) | L (0 , + 12 | s nt ( t ) | s n ( t ) | ˜ u ny ( t ) | L (0 , ≤ | ˜ u nt ( t ) | L (0 , + | s nt ( t ) | (cid:0) C ψ t (˜ u n ( t )) + C (cid:1) ,J ≤ | s nt ( t ) | βs (cid:18) b ∗ | ˜ u n ( t, | + γ | ˜ u n ( t, | (cid:19) ≤ βb ∗ s (cid:18) | s nt ( t ) | | ˜ u n ( t, | (cid:19) + βγ | s nt ( t ) | s | ˜ u n ( t, | ,J ≤ βs | b t ( t ) | ˜ u n ( t, ≤ βs (cid:18) | b t ( t ) | | ˜ u n ( t, | (cid:19) ,J ≤ | s nt ( t ) | s n ( t ) (cid:90) | ˜ u ny ( t ) | dy ≤ | s nt ( t ) | s (cid:0) C ψ t (˜ u n ( t )) + C (cid:1) . For J , by the definition of ψ t , we have that J = | s nt ( t ) | s n ( t ) (cid:32) ψ t (˜ u n ( t )) − s n ( t ) (cid:90) | ˜ u ny ( t, y ) | dy + 1 s n ( t ) (cid:90) ˜ u n ( t, β ( b ( t ) − γξ ) dξ (cid:33) ≤ | s nt ( t ) | s n ( t ) (cid:18) ψ t (˜ u n ( t )) + 1 s n ( t ) βb ∗ ˜ u n ( t, (cid:19) ≤ | s nt ( t ) | s (cid:18) ψ t (˜ u n ( t )) + βb ∗ s n ( t ) (1 + ˜ u n ( t, (cid:19) Hence, by the estimates for each J i and (3.21), we obtain that | ˜ u nt ( t ) | L (0 , + ddt ψ t (˜ u n ( t )) ≤ | s nt ( t ) | s ψ t (˜ u n ( t )) + (cid:18) | s nt ( t ) | + 2 | s nt ( t ) | s (cid:19) ( C ψ t (˜ u n ( t ) + C )+ βb ∗ s | s nt ( t ) | (cid:18) βb ∗ s + βγ | s nt ( t ) | s + β s + | s nt ( t ) | s βb ∗ s (cid:19) ˜ u n ( t, β s | b t ( t ) | + βb ∗ s | s nt ( t ) | s for a.e. t ∈ [0 , T ] . (3.22) free boundary problem describing migration into rubbers – quest of the large time behavior ψ t (˜ u n ( t )) by l ( t ) and otherwise by l ( t ) .Then, by the fact that { s n } is bounded in W , (0 , T ) and (A2), l , l ∈ L (0 , T ) . Now, we see from(3.22) that | ˜ u nt ( t ) | L (0 , + ddt ψ t (˜ u n ( t )) ≤ l ( t ) ψ t (˜ u n ( t )) + l ( t ) for a.e. t ∈ [0 , T ] . Therefore, by using Gronwall’s lemma, we have that (cid:90) t | ˜ u nt ( τ ) | L (0 , dτ + ψ t (˜ u n ( t )) ≤ (cid:20) ψ (˜ u ) + (cid:90) t l ( τ ) dτ (cid:21) e (cid:82) t l ( τ ) dτ for t ∈ [0 , T ] . From this result, we infer that the sequence { ˜ u n } is bounded in W , (0 , T ; L (0 , and the se-quence { ψ ( · ) (˜ u n ( · )) } is bounded in L ∞ (0 , T ) . By these boundedness results and Lemma 3.1, wecan take a sequence { n k } ⊂ { n } such that for some ˜ u ∈ W , (0 , T ; L (0 , ∩ L ∞ (0 , T ; H (0 , , ˜ u n k → ˜ u weakly in W , (0 , T ; L (0 , , weakly -* in L ∞ (0 , T ; H (0 , and in C ( Q ( T )) as k → ∞ . Finally, by letting k → ∞ , we see that ˜ u is a solution of ( AP )(˜ u , s, b ) on [0 , T ] .To complete the proof, we show the uniqueness of a solution ( AP )(˜ u , s, b ) . Let s ∈ W , (0 , T ) with s (0) = s and s ≥ s on [0 , T ] and ˜ u and ˜ u be solutions of ( AP )(˜ u , s, b ) on [0 , T ] . Put ˜ u = ˜ u − ˜ u . Then, by (3.1) and the same argument of the derivation of (3.12), we have that ddt | ˜ u ( t ) | L (0 , + 1 s ( t ) | ˜ u y ( t ) | L (0 , ≤ (cid:90) ys t ( t ) s ( t ) ˜ u y ( t )˜ u ( t ) dy. (3.23)For the right-hand side of (3.23), we deal as follows: (cid:90) ys t ( t ) s ( t ) ˜ u y ( t )˜ u ( t ) dy ≤ s ( t ) | ˜ u y ( t ) | L (0 , + | s t ( t ) | | ˜ u ( t ) | L (0 , . From the above result and (3.23) we obtain that ddt | ˜ u ( t ) | L (0 , + 12 l | ˜ u y ( t ) | L (0 , ≤ | s t ( t ) | | ˜ u ( t ) | L (0 , , where l = max ≤ t ≤ T | s ( t ) | . Therefore, by Gronwall’s lemma we have that | ˜ u ( t ) | L (0 , = 0 for t ∈ [0 , T ] . This implies that ˜ u = ˜ u on [0 , T ] . Thus, Lemma 3.4 is proved. In this section, using the results obtained in Section 3, we establish the existence of a locally-in-timesolution ( PC )(˜ u , s , b ) . In the rest of this section, we assume (A1)-(A3). For T > and l > suchthat s < l we set M ( T, s , l ) := { s ∈ W , (0 , T ) | s ≤ s ≤ l on [0 , T ] , s (0) = s } . Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
Also, for given s ∈ M ( T, s , l ) , we define two solution mappings as follows: Ψ : M ( T, s , l ) → W , (0 , T ; L (0 , ∩ L ∞ (0 , T ; H (0 , by Ψ( s ) = ˜ u , where ˜ u is a unique solution of ( AP )(˜ u , s, b ) ,and Γ T : M ( T, s , l ) → W , (0 , T ) by Γ T ( s ) = s + (cid:82) t a σ (Ψ( s )( τ, dτ for t ∈ [0 , T ] . Moreover,for any K > we put M K ( T ) := { s ∈ M ( T, s , l ) | | s | W , (0 ,T ) ≤ K } . Now, we show that for some
T > , Γ T is a contraction mapping on the closed set of M K ( T ) for any K > . Lemma 4.1.
Let
K > . Then, there exists a positive constant T ∗ ≤ T such that the mapping Γ T ∗ is a contraction on the closed set M K ( T ∗ ) in W , (0 , T ∗ ) . Proof.
For
T > and l > such that s < l , let s ∈ M ( T, s , l ) and ˜ u = Ψ( s ) . First, we note thatit holds | Ψ( s ) | W , (0 ,T ; L (0 , + | Ψ( s ) | L ∞ (0 ,T ; H (0 , ≤ C for s ∈ M K ( T ) , (4.1)where C = C ( T, ˜ u , K, l, b ∗ , β, s ) is a positive constant depending on T , ˜ u , K , l , b ∗ , β and s .Next, we show that there exists T ≤ T such that Γ T : M K ( T ) → M K ( T ) is well-defined. Let K > and s ∈ M K ( T ) . First, by the definition of σ , we see that Γ T ( s )( t ) = s + (cid:90) t a σ (Ψ( s )( τ, dτ ≥ s for t ∈ [0 , T ] . (4.2)Also, by (3.17) and (4.1), we have that | ˜ u ( t, | ≤ √ C e | ˜ u ( t ) | H (0 , ≤ √ C e C for a.e. t ∈ (0 , T ) .Hence, by σ ( r ) ≤ r for r ∈ R we obtain that Γ T ( s )( t ) ≤ s + a (cid:112) C e CT, (cid:90) t | Γ T ( s )( τ ) | dτ ≤ s T + 2 a T ( C e C ) , (4.3)and (cid:90) t | Γ (cid:48) T ( s )( τ ) | dτ ≤ a (cid:90) t | Ψ( s )( τ, | dτ ≤ a T C e C . (4.4)Therefore, by (4.3) and (4.4) we see that there exists T ≤ T such that Γ T ( s ) ∈ M K ( T ) .Next, for s and s ∈ M K ( T ) , let ˜ u = Ψ( s ) and ˜ u = Ψ( s ) and set ˜ u = ˜ u − ˜ u , s = s − s .Then, it holds that ddt | ˜ u ( t ) | H − (cid:90) (cid:18) s ( t ) ˜ u yy ( t ) − s ( t ) ˜ u yy ( t ) (cid:19) ˜ u ( t ) dy = (cid:90) (cid:18) ys t ( t ) s ( t ) ˜ u y ( t ) − ys t ( t ) s ( t ) ˜ u y ( t ) (cid:19) ˜ u ( t ) dy. (4.5) free boundary problem describing migration into rubbers – quest of the large time behavior − (cid:90) (cid:18) s ( t ) ˜ u yy ( t ) − s ( t ) ˜ u yy ( t ) (cid:19) ˜ u ( t ) dy = (cid:90) (cid:18) s ( t ) ˜ u y ( t ) − s ( t ) ˜ u y ( t ) (cid:19) ˜ u y ( t ) dy − (cid:18) s ( t ) ˜ u y ( t, − s ( t ) ˜ u y ( t, (cid:19) ˜ u ( t,
1) + (cid:18) s ( t ) ˜ u y ( t, − s ( t ) ˜ u y ( t, (cid:19) ˜ u ( t, I + I + I . For the term I , the following estimate below holds: I = 1 s ( t ) | ˜ u y ( t ) | L (0 , + (cid:90) (cid:18) s ( t ) − s ( t ) (cid:19) ˜ u y ( t )˜ u y ( t ) dy ≥ s ( t ) | ˜ u y ( t ) | L (0 , − l | s ( t ) | s s ( t ) | ˜ u y ( t ) | L (0 , | ˜ u y ( t ) | L (0 , ≥ (cid:16) − η (cid:17) s ( t ) | ˜ u y ( t ) | L (0 , − η (cid:18) ls (cid:19) | s ( t ) | | ˜ u y ( t ) | L (0 , , where η is arbitrary positive number. For I , we separate in the following way: − (cid:18) s ( t ) ˜ u y ( t, − s ( t ) ˜ u y ( t, (cid:19) ˜ u ( t, a (cid:18) ˜ u ( t, σ (˜ u ( t, s ( t ) − ˜ u ( t, σ (˜ u ( t, s ( t ) (cid:19) ˜ u ( t, a (cid:18) s ( t ) (cid:18) ˜ u ( t, σ (˜ u ( t, − ˜ u ( t, σ (˜ u ( t, (cid:19) + (cid:18) s ( t ) − s ( t ) (cid:19) ˜ u ( t, σ (˜ u ( t, (cid:19) ˜ u ( t, I + I . Similarly to (3.10), the term I is non-positive because the function rσ ( r ) is monotone for r ∈ R .For I , using the fact that σ ( r ) ≤ | r | for r ∈ R and (3.17), the following inequalities hold: | I | = (cid:18) s ( t ) s ( t ) s ( t ) (cid:19) a ˜ u ( t, σ (˜ u ( t, u ( t, ≤ C e ( a ˜ u ( t, s s ( t ) | ˜ u ( t ) | H (0 , | ˜ u ( t ) | L (0 , + 12 | s ( t ) | . (4.6)Put L (1) s ( t ) = C e a | ˜ u ( t, | / s . As for I , we consider the term I as follows: (cid:18) s ( t ) ˜ u y ( t, − s ( t ) ˜ u y ( t, (cid:19) ˜ u ( t, − (cid:18) s ( t ) β ( b ( t ) − γ ˜ u ( t, − s ( t ) β ( b ( t ) − γ ˜ u ( t, (cid:19) ˜ u ( t, − s ( t ) (cid:18) β ( b ( t ) − γ ˜ u ( t, − β ( b ( t ) − γ ˜ u ( t, (cid:19) ˜ u ( t, − (cid:18) s ( t ) − s ( t ) (cid:19) β ( b ( t ) − γ ˜ u ( t, u ( t, . Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
Then, by using (3.17) and (A3), we have that | I | ≤ βC e γs ( t ) | ˜ u ( t ) | H (0 , | ˜ u ( t ) | L (0 , + ( β ( b ∗ + γ | ˜ u ( t, | ) C e s s ( t ) | ˜ u ( t ) | H (0 , | ˜ u ( t ) | L (0 , + 12 | s ( t ) | for t ∈ [0 , T ] . (4.7)For the right-hand side of (4.5), we can write as follows: (cid:90) (cid:18) ys t ( t ) s ( t ) ˜ u y ( t ) − ys t ( t ) s ( t ) ˜ u y ( t ) (cid:19) ˜ u ( t ) dy = (cid:90) ys t ( t ) s ( t ) ˜ u y ( t )˜ u ( t ) dy + (cid:90) ys t ( t ) s ( t ) ˜ u y ( t )˜ u ( t ) dy + (cid:90) (cid:18) s ( t ) − s ( t ) (cid:19) ys t ( t )˜ u y ( t )˜ u ( t ) dy := I + I + I . The three terms are estimated in the following way: I ≤ η s ( t ) | ˜ u y ( t ) | L (0 , + 12 η | s t ( t ) | | ˜ u ( t ) | L (0 , ,I ≤ s (cid:18) | s t ( t ) | + | ˜ u y ( t ) | L (0 , | ˜ u ( t ) | L (0 , (cid:19) ,I ≤ s (cid:18) | s ( t ) | | ˜ u y ( t ) | L (0 , + | s t ( t ) | | ˜ u ( t ) | L (0 , (cid:19) . Then, by (4.5)-(4.7) we obtain that ddt | ˜ u ( t ) | L (0 , + (1 − η ) 1 s ( t ) | ˜ u y ( t ) | L (0 , ≤ βC e γs ( t ) | ˜ u ( t ) | H (0 , | ˜ u ( t ) | L (0 , + 1 s ( t ) (cid:18) L (1) s ( t ) + ( β ( b ∗ + γ | ˜ u ( t, | ) C e s (cid:19) | ˜ u ( t ) | H (0 , | ˜ u ( t ) | L (0 , + (cid:18) η | s t ( t ) | + 12 s | ˜ u y ( t ) | L (0 , + 12 s | s t ( t ) | (cid:19) | ˜ u ( t ) | L (0 , + (cid:32) s | ˜ u y ( t ) | L (0 , + 12 η (cid:18) ls (cid:19) | ˜ u y | L (0 , + 1 (cid:33) | s ( t ) | + 12 s | s t ( t ) | . (4.8)Here, by (3.17) and (4.1), we see that | ˜ u i ( t, | ≤ C e ( | ˜ u iy ( t ) | L (0 , | ˜ u i ( t ) | L (0 , + | ˜ u i ( t ) | L (0 , ) ≤ C e C for t ∈ [0 , T ] , (4.9)where C is the same constant as in (4.1). Then, by (4.9) we note that { L (1) s | s ∈ M k ( T ) } is bounded free boundary problem describing migration into rubbers – quest of the large time behavior L ∞ (0 , T ) . Also, by putting C = ( β ( b ∗ + 2 γC e C )) C e and Young’s inequality it follows that βC e γs ( t ) | ˜ u ( t ) | H (0 , | ˜ u ( t ) | L (0 , ≤ βC e γs ( t ) (cid:18) | ˜ u y ( t ) | L (0 , | ˜ u ( t ) | L (0 , + | ˜ u ( t ) | L (0 , (cid:19) ≤ βC e γ (cid:18) η s ( t ) | ˜ u y ( t ) | L (0 , + ( 12 η + 1 s ) | ˜ u ( t ) | L (0 , (cid:19) , and (cid:18) L (1) s ( t ) + C s (cid:19) s ( t ) | ˜ u ( t ) | H (0 , | ˜ u ( t ) | L (0 , ≤ (cid:18) L (1) s ( t ) + C s (cid:19) s ( t ) ( | ˜ u y ( t ) | L (0 , | ˜ u ( t ) | L (0 , + | ˜ u ( t ) | L (0 , ) ≤ (cid:18) L (1) s ( t ) + C s (cid:19) (cid:20) s ( t ) η | ˜ u y ( t ) | L (0 , + 1 s ( 12 η + 1) | ˜ u ( t ) | L (0 , (cid:21) . Accordingly, by applying these results to (4.8) and taking a suitable η = η , we have ddt | ˜ u ( t ) | L (0 , + 12 1 s ( t ) | ˜ u y ( t ) | L (0 , ≤ βC e γ (cid:18) η + 1 s (cid:19) | ˜ u ( t ) | L (0 , + (cid:18) L (1) s ( t ) + C s (cid:19) s (cid:18) η + 1 (cid:19) | ˜ u ( t ) | L (0 , + (cid:18) η | s t ( t ) | + 12 s | ˜ u y ( t ) | L (0 , + 12 s | s t ( t ) | (cid:19) | ˜ u ( t ) | L (0 , + (cid:32) s | ˜ u y ( t ) | L (0 , + 12 η (cid:18) ls (cid:19) | ˜ u y ( t ) | L (0 , + 1 (cid:33) | s ( t ) | + 12 s | s t ( t ) | . (4.10)Now, we put the summation of all coefficients of | ˜ u ( t ) | L (0 , by L (2) s ( t ) for t ∈ [0 , T ] and L (3) s ( t ) = | ˜ u y ( t ) | L (0 , / s + (4 l | ˜ u y ( t ) | L (0 , ) / η s + 1 + 1 / s . Then, we have ddt | ˜ u ( t ) | L (0 , + 12 1 s ( t ) | ˜ u y ( τ ) | L (0 , ≤ L (2) s ( t ) | ˜ u ( t ) | L (0 , + L (3) s ( t )( | s ( t ) | + | s t ( t ) | ) for t ∈ [0 , T ] . (4.11)Here, using (4.1) and the fact that s i ∈ M K ( T ) for i = 1 , , we see that L (2) s ∈ L (0 , T ) and L (3) s ∈ L ∞ (0 , T ) . Therefore, Gronwall’s inequality guarantees that | ˜ u ( t ) | L (0 , + 12 1 s ( t ) (cid:90) t | ˜ u y ( τ ) | L (0 , dτ ≤ (cid:16) | L (3) s | L ∞ (0 ,T ) | s | W , (0 ,T ) (cid:17) e (cid:82) t L (2) s ( τ ) dτ for t ∈ [0 , T ] . (4.12)8 Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
By using (4.12) we show that there exists T ∗ ≤ T such that Γ T ∗ is a contraction mapping on theclosed subset of M K ( T ∗ ) . To do so, from the subtraction of the time derivatives of Γ T ( s ) and Γ T ( s ) and relying on (3.17) and (4.12), we have for T ≤ T the following estimate: | (Γ T ( s )) t − (Γ T ( s )) t | L (0 ,T ) ≤ a (cid:18) | σ (˜ u ( · , − σ (˜ u ( · , | L (0 ,T ) (cid:19) ≤ a (cid:112) C e (cid:18)(cid:90) T ( | ˜ u y ( t ) | L (0 , | ˜ u ( t ) | L (0 , + | ˜ u ( t ) | L (0 , ) dt (cid:19) / ≤ a (cid:112) C e (cid:32) | ˜ u | L ∞ (0 ,T ; L (0 , (cid:18)(cid:90) T | ˜ u y ( t ) | L (0 , dt (cid:19) + (cid:112) T | ˜ u | L ∞ (0 ,T ; L (0 , (cid:33) . (4.13)Using (4.12), we obtain | Γ T ( s ) − Γ T ( s ) | W , (0 ,T ) ≤ T C (cid:18) T | s | W , (0 ,T ) + (cid:112) T | s | W , (0 ,T ) (cid:19) , (4.14)where C is a positive constant obtained by (4.12). Therefore, by (4.13) and (4.14) we see that thereexists T ∗ ≤ T such that Γ T ∗ is a contraction mapping on a closed subset of M K ( T ∗ ) .From Lemma 4.1, by applying Banach’s fixed point theorem, there exists s ∈ M K ( T ∗ ) , where T ∗ is the same as in Lemma 4.1 such that Γ T ∗ ( s ) = s . This implies that ( PC )(˜ u , s , b ) has a uniquesolution ( s, ˜ u ) on [0 , T ∗ ] . Thus, we can prove Theorem 2.4. Moreover, this shows that by the changeof variables (2.8) a pair of the function ( s, u ) is a solution of ( P )( u , s , b ) on [0 , T ∗ ] .At the end of this section, we show the boundedness of the solution to ( P )( u , s , b ) . Lemma 4.2.
Let
T > and ( s, u ) be a solution of ( P )( u , s , b ) on [0 , T ] . Then, ≤ u ( t ) ≤ b ∗ /γ on [0 , s ( t )] for t ∈ [0 , T ] . Proof.
First, we show that u ( t ) ≥ on [0 , s ( t )] for t ∈ [0 , T ] . From (1.1), we have that ddt (cid:90) s ( t )0 | [ − u ( t )] + | dz − s t ( t )2 | [ − u ( t, s ( t ))] + | + (cid:90) s ( t )0 u zz ( t )[ − u ( t )] + dz = 0 for a.e. t ∈ [0 , T ] . (4.15)By the boundary conditions (1.2) and (1.3) it follows that u z ( t, s ( t ))[ − u ( t, s ( t ))] + = − u ( t, s ( t )) s t ( t )[ − u ( t, s ( t ))] + = s t ( t ) | [ − u ( t, s ( t ))] + | and − u z ( t, − u ( t, + = β ( b ( t ) − γu ( t, − u ( t, + ≥ . free boundary problem describing migration into rubbers – quest of the large time behavior ddt (cid:90) s ( t )0 | [ − u ( t )] + | dz + s t ( t )2 | [ − u ( t, s ( t ))] + | + (cid:90) s ( t )0 | [ − u ( t )] + z | dz ≤ for a.e. t ∈ [0 , T ] . (4.16)Note that by s t ( t ) = a σ ( u ( t, s ( t )) , the second term in the left-hand side of (4.16) is equal to 0.Therefore, by integrating (4.16) over [0 , T ] we conclude that u ≥ on [0 , s ( t )] for t ∈ [0 , T ] .Next, we show that u ( t ) ≤ b ∗ /γ on [0 , s ( t )] for t ∈ [0 , T ] . Put U ( t, z ) = [ u ( t, z ) − b ∗ /γ ] + for z ∈ [0 , s ( t )] and t ∈ [0 , T ] . Then, we have that ddt (cid:90) s ( t )0 | U ( t, z ) | dz − s t ( t )2 | U ( t, s ( t )) | − (cid:90) s ( t )0 u zz ( t ) U ( t, z ) dz = 0 for a.e. t ∈ [0 , T ] . (4.17)Using the boundary condition (1.2), it holds that − u z ( t, s ( t )) U ( t, s ( t )) = u ( t, s ( t )) s t ( t ) U ( t, s ( t ))= s t ( t ) (cid:18) u ( t, s ( t )) − b ∗ γ (cid:19) U ( t, s ( t )) + s t ( t ) b ∗ γ U ( t, s ( t ))= s t ( t ) | U ( t, s ( t )) | + s t ( t ) b ∗ γ U ( t, s ( t )) . Also, by (1.3) and b ≤ b ∗ , we observe that u z ( t, U ( t,
0) = − β ( b ( t ) − γu ( t, U ( t, β ( γu ( t, − b ∗ + b ∗ − b ( t )) U ( t, βγ | U ( t, | + β ( b ∗ − b ( t )) U ( t, ≥ . By applying the above two results to (4.17) we obtain that ddt (cid:90) s ( t )0 | U ( t, z ) | dz + (cid:90) s ( t )0 | U z ( t, z ) | dz + s t ( t )2 | U ( t, s ( t )) | + s t ( t ) b ∗ γ U ( t, s ( t )) ≤ for a.e. t ∈ [0 , T ] . (4.18)Here, by s t ( t ) = a σ ( u ( t, s ( t ))) we notice that s t ( t ) ≥ on [0 , T ] , and the third and forth terms inthe left-hand side of (4.18) are non-negative. Therefore, we have that ddt (cid:90) s ( t )0 | U ( t, z ) | dz + (cid:90) s ( t )0 | U z ( t, z ) | dz ≤ for a.e. t ∈ [0 , T ] . (4.19)Finally, by integrating (4.19) over [0 , t ] for t ∈ [0 , T ] and using (A3), we see that u ( t ) ≤ b ∗ /γ on [0 , s ( t )] for t ∈ [0 , T ] . Thus, Lemma 4.2 is proven.By Lemma 4.2, we can conclude that Theorem 2.2 holds.0 Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
In this section, we prove Theorem 2.5 which ensure the existence and uniqueness of a globally-in-time solution of (P) ( u , s , b ) . First, we provide uniform estimates of a solution of (P) ( u , s , b ) . Lemma 5.1.
Let ( s, u ) be a solution of (P) ( u , s , b ) on [0 , T ] satisfying ≤ u ≤ b ∗ /γ on [0 , s ( t )] for t ∈ [0 , T ] . Then, there exists a positive constant ˜ C which is independent of T such that (cid:90) t | u t ( τ ) | L (0 ,s ( τ )) dτ + | u z ( t ) | L (0 ,s ( t )) ≤ ˜ C for all t ∈ (0 , T ) . (5.1) Proof.
Let ( s, u ) be a solution of (P) ( u , s , b ) on [0 , T ] such that ≤ u ≤ b ∗ /γ on Q s ( T ) . Then,by the change of variables (2.1) we see that ( s, ˜ u ) is a solution of (PC) (˜ u , s , b ) on [0 , T ] in thesense of Definition 2.3 and satisfies that ≤ ˜ u ≤ b ∗ /γ on Q ( T ) . Now, we put v h ( t ) = ˜ u ( t ) − ˜ u ( t − h ) h for h > and u ( t ) = u (0) = u and b ( t ) = b (0) for t < . By (1.1), it holds that (cid:90) ˜ u t ( t ) s ( t ) v h ( t ) dy − (cid:90) s ( t ) ˜ u yy ( t ) v h ( t ) dy = (cid:90) ys t ( t )˜ u y ( t ) s ( t ) s ( t ) v h ( t ) dy for t ∈ [0 , T ] . (5.2)Then, using (1.2)-(1.4) and s t ( t ) = a σ (˜ u ( t, a ˜ u ( t, for t ∈ [0 , T ] , we observe that − (cid:90) s ( t ) ˜ u yy ( t ) ˜ u ( t ) − ˜ u ( t − h ) h dy = a ˜ u ( t, v h ( t, − β ( b ( t ) − γ ˜ u ( t, v h ( t,
0) + (cid:90) s ( t ) ˜ u y ( t ) v hy ( t ) dy, (5.3)and (cid:90) s ( t ) ˜ u y ( t ) v hy ( t ) dy ≥ h (cid:90) s ( t ) ( | ˜ u y ( t ) | − | ˜ u y ( t − h ) | ) dy = 12 h (cid:20)(cid:90) s ( t )0 | u z ( t ) | dz − (cid:90) s ( t − h )0 s ( t − h ) s ( t ) | u z ( t − h ) | dz (cid:21) = 12 h (cid:20)(cid:90) s ( t )0 | u z ( t ) | dz − (cid:90) s ( t − h )0 | u z ( t − h ) | dz + (cid:90) s ( t − h )0 s ( t ) − s ( t − h ) s ( t ) | u z ( t − h ) | dz (cid:21) . (5.4)Here, for t ∈ [0 , T ] the following inequality holds: a ˜ u ( t, v h ( t,
1) = a ˜ u ( t, − ˜ u ( t, u ( t − h, h ≥ a ˜ u ( t, − ˜ u ( t − h, h . (5.5) free boundary problem describing migration into rubbers – quest of the large time behavior Φ( b ( t ) , r ) = − β ( b ( t ) r − γ r ) for r ∈ R , it is easy to see that ∂ ∂r Φ( b ( t ) , r ) = βγ ≥ for r ∈ R . Hence, for t ∈ [0 , T ] , Φ( b ( t ) , ˜ u ( t, is convex with respect to the secondcomponent so that we can see that the following inequality holds. − β ( b ( t ) − γ ˜ u ( t, v h ( t, ≥ Φ( b ( t ) , ˜ u ( t, − Φ( b ( t ) , ˜ u ( t − h, h for t ∈ [0 , T ] . (5.6)Combining (5.2)-(5.6) with (5.1) , we have (cid:90) ˜ u t ( t ) s ( t ) v h ( t ) dy + 12 h (cid:20)(cid:90) s ( t )0 | u z ( t ) | dz − (cid:90) s ( t − h )0 | u z ( t − h ) | dz + (cid:90) s ( t − h )0 s ( t ) − s ( t − h ) s ( t ) | u z ( t − h ) | dz (cid:21) + a ˜ u ( t, − ˜ u ( t − h, h + Φ( b ( t ) , ˜ u ( t, − Φ( b ( t ) , ˜ u ( t − h, h ≤ (cid:90) ys t ( t )˜ u y ( t ) v h ( t ) dy for t ∈ [0 , T ] . (5.7)Now, we integrate (5.7) over [0 , t ] for t ∈ (0 , T ] and take the limit as h → . Then, by the changeof variables (2.8) the first term of the left-hand side of (5.7) is as follows: lim h → (cid:90) t (cid:90) ˜ u t ( t ) s ( t ) v h ( t ) dydt = (cid:90) t (cid:90) | ˜ u t ( t ) | s ( t ) dydt = (cid:90) t (cid:90) s ( t )0 (cid:18) | u t ( t ) | + 2 u t ( t ) u z ( t ) zs ( t ) s t ( t ) + (cid:18) u z ( t ) zs ( t ) s t ( t ) (cid:19) (cid:19) dzdt. (5.8)As arguing the local existence, the function t → (cid:82) s ( t )0 | u z ( t ) | dz is absolutely continuous on [0 , T ] .Then, the second and third terms of the left-hand side of (5.7) can be dealt with as lim h → h (cid:90) t (cid:18)(cid:90) s ( t )0 | u z ( t ) | dz − (cid:90) s ( t − h )0 | u z ( t − h ) | dz (cid:19) dτ = 12 (cid:18)(cid:90) s ( t )0 | u z ( t ) | dz − (cid:90) s | u z (0) | dz (cid:19) , (5.9)and lim h → (cid:90) t (cid:90) s ( t − h )0 s ( t ) s ( t ) − s ( t − h ) h | u z ( t − h ) | dzdt = 12 (cid:90) t (cid:90) s ( t )0 s t ( t ) s ( t ) | u z ( t ) | dzdt. (5.10)Moreover, since ˜ u is continuous on Q ( T ) we have that lim h → a h (cid:90) t (cid:18) ˜ u ( t, − ˜ u ( t − h, (cid:19) dt = lim h → (cid:18) a h (cid:90) t t − h ˜ u ( t, dt (cid:19) − a u (1)= a u ( t , − a u (1) . (5.11)2 Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
Similarly to the derivation of (5.11), lim h → h (cid:90) t (cid:18) Φ( b ( t ) , ˜ u ( t, − Φ( b ( t ) , ˜ u ( t − h, (cid:19) dt =Φ( b ( t ) , ˜ u ( t , − Φ( b (0) , ˜ u (0))+ lim h → (cid:18) − h (cid:90) t (cid:20) Φ( b ( t ) , ˜ u ( t − h, − Φ( b ( t − h ) , ˜ u ( t − h, (cid:21) dt (cid:19) . (5.12)For the last term of the right-hand side of (5.12) we observe that lim h → (cid:18) − h (cid:90) t (cid:20) Φ( b ( t ) , ˜ u ( t − h, − Φ( b ( t − h ) , ˜ u ( t − h, (cid:21) dt (cid:19) ≥ lim h → (cid:18) − h (cid:90) t β | b ( t ) − b ( t − h ) || ˜ u ( t − h, | dt (cid:19) ≥ lim h → (cid:18) − βh (cid:90) t (cid:18)(cid:90) tt − h | b t ( τ ) | dτ (cid:19) | ˜ u ( t − h, | dt (cid:19) ≥ − βb ∗ γ (cid:90) t | b t ( t ) | dt. (5.13)From (5.7) and the estimates (5.8)-(5.13), we obtain that (cid:90) t (cid:90) s ( t )0 (cid:18) | u t ( t ) | + 2 u t ( t ) u z ( t ) zs ( t ) s t ( t ) + (cid:18) u z ( t ) zs ( t ) s t ( t ) (cid:19) (cid:19) dzdt + 12 (cid:90) s ( t )0 | u z ( t ) | dz − (cid:90) s | u z (0) | dz + 12 (cid:90) t (cid:90) s ( t )0 s t ( t ) s ( t ) | u z ( t ) | dzdt + a u ( t , − a u (1) + Φ( b ( t ) , ˜ u ( t , − Φ( b (0) , ˜ u (0)) − βb ∗ γ (cid:90) t | b t ( t ) | dt ≤ (cid:90) t (cid:90) s ( t )0 (cid:18) u t ( t ) u z ( t ) zs ( t ) s t ( t ) + (cid:18) u z ( t ) zs ( t ) s t ( t ) (cid:19) (cid:19) dzdt for t ∈ [0 , T ] . (5.14)Then, we see that the third term of the left-hand side of (5.14) is same to the second term of theright-hand side of (5.14) . Then, by moving the second term of the left-hand side of (5.14) to theright-hand side we have that (cid:90) t (cid:90) s ( t )0 | u t ( t ) | dzdt + 12 (cid:90) s ( t )0 | u z ( t ) | dz − (cid:90) s | u z (0) | dz + 12 (cid:90) t (cid:90) s ( t )0 s t ( t ) s ( t ) | u z ( t ) | dzdt + a u ( t , − a u (1) + Φ( b ( t ) , ˜ u ( t , − Φ( b (0) , ˜ u (0)) − βb ∗ γ (cid:90) t | b t ( t ) | dt ≤ (cid:90) t (cid:90) s ( t )0 − u t ( t ) u z ( t ) zs ( t ) s t ( t ) dzdt for t ∈ [0 , T ] . (5.15) free boundary problem describing migration into rubbers – quest of the large time behavior s t ( t ) ≥ for t ∈ [0 , T ] we obtain the following inequality: − (cid:90) t (cid:90) s ( t )0 u t ( t ) u z ( t ) zs ( t ) s t ( t ) dzdt = − (cid:90) t (cid:90) s ( t )0 u zz ( t ) u z ( t ) zs ( t ) s t ( t ) dzdt = − (cid:90) t (cid:90) s ( t )0 (cid:18) ∂∂z | u z ( t ) | (cid:19) zs ( t ) s t ( t ) dzdt = − (cid:90) t | u z ( t, s ( t )) | s t ( t ) dt + 12 (cid:90) t (cid:90) s ( t )0 s t ( t ) s ( t ) | u z ( t ) | dzdt. ≤ (cid:90) t (cid:90) s ( t )0 s t ( t ) s ( t ) | u z ( t ) | dzdt. (5.16)Hence, by (5.15) and (5.16) we have that (cid:90) t (cid:90) s ( t )0 | u t ( t ) | dzdt + 12 (cid:90) s ( t )0 | u z ( t ) | dz − (cid:90) s | u z (0) | dz + 12 (cid:90) t (cid:90) s ( t )0 s t ( t ) s ( t ) | u z ( t ) | dzdt + a u ( t , − a u (1) + Φ( b ( t ) , ˜ u ( t , − Φ( b (0) , ˜ u (0)) − βb ∗ γ (cid:90) t | b t ( t ) | dt ≤ (cid:90) t (cid:90) s ( t )0 s t ( t ) s ( t ) | u z ( t ) | dzdt for t ∈ [0 , T ] . (5.17)The forth term in the left-hand side and the right-hand side are canceled out and the fifth and eighthterms in the left-hand side are positive. Therefore, we finally obtain that (cid:90) t (cid:90) s ( t )0 | u t ( t ) | + 12 (cid:90) s ( t )0 | u z ( t ) | dz ≤ (cid:90) s | u z (0) | dz + a u (1) − Φ( b ( t ) , ˜ u ( t , βb ∗ γ (cid:90) t | b t ( t ) | dt for t ∈ [0 , T ] . (5.18)In the right-hand side of (5.18), by (A2) and ≤ ˜ u ( t ) ≤ b ∗ /γ on [0 , for t ∈ [0 , T ] and thedefinition of Φ , we can estimate as follows: − Φ( b ( t ) , ˜ u ( t , − βb ( t )˜ u ( t ,
0) + βγ u ( t , ≤ βγ (cid:18) b ∗ γ (cid:19) . (5.19)Finally, by (5.19), b ∈ W , (0 , T ) as in (A2) and (A3) we see that there exists ˜ C which depends on b ∗ , a , γ , β such that (5.1) holds. Thus, Lemma 5.1 is proved.At the end of this section, we prove Theorem 2.5. Let T > . By the local existence result thereexists T < T such that (P) ( u , s , b ) has a unique solution ( s, u ) on [0 , T ] satisfying ≤ u ≤ b ∗ /γ Kota Kumazaki, Toyohiko Aiki and Adrian Muntean on Q s ( T ) . Then, the pair ( s, ˜ u ) with the variable (2.1) is a solution of ( PC )(˜ u , s , b ) satisfying ≤ ˜ u ≤ b ∗ /γ on Q ( T ) . Let put ˜ T := sup { T > | ( PC )(˜ u , s , b ) has a solution ( s, ˜ u ) on [0 , T ] } . From the local existence result, we deduce that ˜ T > . Now, we assume ˜ T < T . First, by (2.5) andthe result that ˜ u ( t ) ≥ on [0 , for t ∈ [0 , ˜ T ) we see that s t ( t ) ≥ for t ∈ [0 , ˜ T ) , and therefore s ( t ) ≥ s for t ∈ [0 , ˜ T ) . Also, by putting L ( t ) = a b ∗ γ t + s for t ∈ [0 , T ] we have that s ( t ) = s + (cid:90) t a σ (˜ u ( τ, dτ = s + (cid:90) t a ˜ u ( τ, dτ ≤ s + a b ∗ γ ˜ T = L ( ˜ T ) < L ( T ) for t ∈ [0 , ˜ T ) . (5.20)Next, by using the change of the variable (2.1), it holds that (cid:90) | ˜ u y ( t ) | dy = (cid:90) s ( t )0 s ( t ) | u z ( t ) s ( t ) | dzdt. Therefore, from (5.20) and Lemma 5.1, we obtain that | ˜ u y ( t ) | L (0 , ≤ L ( T ) ˜ C for all t < ˜ T , (5.21)where ˜ C is the same constant as in Lemma 5.1. By (5.21) we see that for some ˜ u ˜ T ∈ H (0 , , ˜ u ( t ) → ˜ u ˜ T strongly in L (0 , and weakly in H (0 , as t → ˜ T and ≤ ˜ u ˜ T ≤ b ∗ /γ on (0 , .Also, by | s t ( t ) | ≤ a b ∗ /γ for t ∈ [0 , ˜ T ) , { s ( t ) } t ∈ [0 , ˜ T ) is a Cauchy sequence in R so that for some s ˜ T ∈ R , s ( t ) → s ˜ T in R as t → ˜ T . Moreover, by (5.20), s ˜ T satisfies that < s ≤ s ˜ T ≤ L ( ˜ T ) .Now, we put u ˜ T ( z ) = ˜ u ˜ T ( zs ˜ T ) for z ∈ [0 , s ˜ T ] . Then, we see that u ˜ T ∈ H (0 , s ˜ T ) and ≤ u ˜ T ≤ b ∗ /γ on (0 , s ˜ T ) and we can consider ( s ˜ T , u ˜ T ) as a initial data. Therefore, by repeating the argument ofthe local existence we can extend a solution beyond ˜ T . This is a contradiction for the definition of ˜ T and we have a solution on the whole interval [0 , T ] . Thus Theorem 2.5 is proved. In this section, we discuss the large time behavior of a solution to (P) ( u , s , b ) as t → ∞ . First, weassume (A2)’ replaced by (A2):(A2)’: b ∈ W , loc ([0 , ∞ )) , b t ∈ L (0 , ∞ ) , lim t →∞ b ( t ) = b ∞ , b − b ∞ ∈ L (0 , ∞ ) and b ∗ ≤ b ≤ b ∗ on (0 , ∞ ) , where b ∗ and b ∗ are positive constants as in (A2).Clearly, we see that b ∗ ≤ b ∞ ≤ b ∗ . Next, we consider the following stationary problem (P) ∞ : find a free boundary problem describing migration into rubbers – quest of the large time behavior ( u ∞ , s ∞ ) ∈ L (0 , s ∞ ) × R satisfying − u ∞ zz = 0 on (0 , s ∞ ) , − u ∞ z (0) = β ( b ∞ − γu ∞ (0)) , − u ∞ z ( s ∞ ) = 0 ,u ∞ ( s ∞ ) = 0 . By using the change of variables ˜ u ∞ ( y ) = u ∞ ( ys ∞ ) for y ∈ (0 , , (P) ∞ can be written in thefollowing problem (˜ P ) ∞ : − s ∞ ˜ u ∞ yy = 0 on (0 , , − s ∞ ˜ u ∞ y (0) = β ( b ∞ − γ ˜ u ∞ (0)) , − s ∞ ˜ u ∞ y (1) = 0 , ˜ u ∞ (1) = 0 . The next lemma is concerned with non-existence of a solution ( s ∞ , ˜ u ∞ ) of the problem (˜ P ) ∞ . Lemma 6.1.
A solution ( s ∞ , ˜ u ∞ ) of (˜ P ) ∞ satisfying < s ∞ < + ∞ and ˜ u ∞ ∈ H (0 , does notexist. Proof.
Let ( s ∞ , ˜ u ∞ ) be a solution of (˜ P ) ∞ such that < s ∞ < + ∞ and ˜ u ∞ ∈ H (0 , . Then, itholds that − s ∞ ˜ u ∞ y (1) + 1 s ∞ ˜ u ∞ y (0) = 0 . Then, we see that ˜ u ∞ (0) = b ∞ /γ . Hence, ˜ u ∞ ∈ H (0 , satisfies − ˜ u ∞ yy = 0 on (0 , with ˜ u ∞ y (1) = ˜ u ∞ y (0) = 0 and ˜ u ∞ (1) = 0 so that ˜ u ∞ ≡ on [0 , . This is a contradiction to ˜ u ∞ (0) (cid:54) = 0 .Thus, we conclude that Lemma 6.1 holds.Now, we state the result on the large time behavior of a solution as t → ∞ . Theorem 6.2.
Assume (A1), (A2)’ and (A3) and let (P) ( u , s , b ) be a solution ( s, u ) on [0 , ∞ ) .Then, s → ∞ as t → ∞ .We prove this result in the rest of the section. To prove Theorem 6.2, we provide some uniform estimates for the solution with respect to time t .We assume (A1), (A2)’ and (A3). Then, by Theorem 2.5, (P) ( u , s , b ) has a solution ( s, u ) on [0 , T ] for T > satisfying ≤ u ≤ b ∗ /γ on [0 , s ( t )] for t ∈ [0 , T ] .6 Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
Lemma 6.3.
Let ( s, u ) be a solution of (P) ( u , s , b ) on [0 , ∞ ) . If there exists a constant C > such that s ( t ) ≤ C for t > , then it holds ( i ) (cid:90) t | s t ( τ ) | dτ + (cid:90) t | u z ( τ ) | L (0 ,s ( τ )) dτ + (cid:90) t (cid:12)(cid:12)(cid:12)(cid:12) u ( τ, − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) dτ ≤ C for t > , (6.1) ( ii ) (cid:90) t | u t ( τ ) | L (0 ,s ( τ )) dτ + | u z ( t ) | L (0 ,s ( t )) ≤ C for t > , (6.2)where C and C are positive constants which is independent of time t . Proof.
First, we prove that (6.1) holds. By (1.1) we have that ddt (cid:90) s ( t )0 (cid:12)(cid:12)(cid:12)(cid:12) u ( t ) − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) dz − s t ( t )2 (cid:12)(cid:12)(cid:12)(cid:12) u ( t, s ( t )) − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) − (cid:90) s ( t )0 u zz ( t ) (cid:18) u ( t ) − b ∞ γ (cid:19) dz = 0 . (6.3)For the third term of the left-hand side of (6.3), it holds that − (cid:90) s ( t )0 u zz ( t ) (cid:18) u ( t ) − b ∞ γ (cid:19) = − u z ( t, s ( t )) (cid:18) u ( t, s ( t )) − b ∞ γ (cid:19) + u z ( t, (cid:18) u ( t, − b ∞ γ (cid:19) + (cid:90) s ( t )0 | u z ( t ) | dz = s t ( t ) (cid:12)(cid:12)(cid:12)(cid:12) u ( t, s ( t )) − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) + b ∞ γ s t ( t ) (cid:18) u ( t, s ( t )) − b ∞ γ (cid:19) − β ( b ( t ) − γu ( t, (cid:18) u (0) − b ∞ γ (cid:19) + (cid:90) s ( t )0 | u z ( t ) | dz. (6.4)By (6.3) with (6.4) it follows that ddt (cid:90) s ( t )0 (cid:12)(cid:12)(cid:12)(cid:12) u ( t ) − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) dz + s t ( t )2 (cid:12)(cid:12)(cid:12)(cid:12) u ( t, s ( t )) − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:90) s ( t )0 | u z ( t ) | dz + b ∞ γ s t ( t ) (cid:18) u ( t, s ( t )) − b ∞ γ (cid:19) − β ( b ( t ) − γu ( t, (cid:18) u (0) − b ∞ γ (cid:19) = 0 . (6.5)Since s t ( t ) = a σ ( u ( t, s ( t )) = a u ( t, s ( t )) , we have that b ∞ γ s t ( t ) (cid:18) u ( t, s ( t )) − b ∞ γ (cid:19) = b ∞ γ | s t ( t ) | a − (cid:18) b ∞ γ (cid:19) s t ( t ) . (6.6)Also, it holds that − β ( b ( t ) − γu ( t, (cid:18) u (0) − b ∞ γ (cid:19) = β ( γu ( t, − b ∞ + b ∞ − b ( t )) (cid:18) u ( t, − b ∞ γ (cid:19) = βγ (cid:12)(cid:12)(cid:12)(cid:12) u ( t, − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) + β ( b ∞ − b ( t )) (cid:18) u ( t, − b ∞ γ (cid:19) . (6.7) free boundary problem describing migration into rubbers – quest of the large time behavior ddt (cid:90) s ( t )0 (cid:12)(cid:12)(cid:12)(cid:12) u ( t ) − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) dz + s t ( t )2 (cid:12)(cid:12)(cid:12)(cid:12) u ( t, s ( t )) − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:90) s ( t )0 | u z ( t ) | dz + b ∞ γ | s t ( t ) | a + βγ (cid:12)(cid:12)(cid:12)(cid:12) u ( t, − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) b ∞ γ (cid:19) s t ( t ) + β ( b ( t ) − b ∞ ) (cid:18) u ( t, − b ∞ γ (cid:19) . (6.8)Here, by the fact that u ( t ) ≥ on [0 , s ( t )] for t ∈ [0 , T ] we note that s t ( t ) ≥ for t ∈ [0 , T ] and thesecond term of the left-hand side of (6.8) is non-negative. Hence, we derive that ddt (cid:90) s ( t )0 (cid:12)(cid:12)(cid:12)(cid:12) u ( t ) − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) dz + (cid:90) s ( t )0 | u z ( t ) | dz + b ∞ γ | s t ( t ) | a + βγ (cid:12)(cid:12)(cid:12)(cid:12) u ( t, − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) b ∞ γ (cid:19) s t ( t ) + β ( b ( t ) − b ∞ ) (cid:18) u ( t, − b ∞ γ (cid:19) . (6.9)By using u ( t ) ≤ b ∗ /γ on [0 , s ( t )] for t ∈ [0 , T ] and integrating over [0 , t ] for t ∈ [0 , T ] we obtainthat (cid:90) s ( t )0 (cid:12)(cid:12)(cid:12)(cid:12) u ( t ) − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) dz + (cid:90) t (cid:90) s ( t )0 | u z ( t ) | dzdt + b ∞ a γ (cid:90) t | s t ( t ) | dt + βγ (cid:90) t (cid:12)(cid:12)(cid:12)(cid:12) u ( t, − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ (cid:90) s (cid:12)(cid:12)(cid:12)(cid:12) u − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) dz + (cid:18) b ∞ γ (cid:19) ( s ( t ) − s (0)) + β ( b ∗ + b ∞ ) γ (cid:90) t | b ∞ − b ( t ) | dt. (6.10)Hence, from b − b ∞ ∈ L (0 , ∞ ) in (A2)’ and the assumption that s ( t ) ≤ C for t ∈ [0 , T ] , weconclude that (6.1) holds.Also, for the estimate (6.2), by repeating the proof of Lemma 5.1 we infer that it holds that (cid:90) t (cid:90) s ( t )0 | u t ( t ) | + 12 (cid:90) s ( t )0 | u z ( t ) | dz ≤ (cid:90) s | u z (0) | dz + | ˜ u (1) | + | Φ( b ( t ) , ˜ u ( t , | + βb ∗ γ (cid:90) t | b t ( t ) | dt for t ∈ [0 , T ] . (6.11)Therefore, by (5.19) and b t ∈ L (0 , ∞ ) we can find a positive constant C which is independent of t such that (6.2) holds. This completes the proof of this lemma. At the end of the paper, by using the uniform estimate obtained in previous subsection, we completethe proof of Theorem 6.2 concerning the large-time behavior of solutions to (P) ( u , s , b ) .8 Kota Kumazaki, Toyohiko Aiki and Adrian Muntean
Let us assume (A1), (A2)’ and (A3). Then, by Theorem 2.5, we have a solution ( s, u ) of(P) ( u , s , b ) on [0 , T ] for any T > such that ≤ u ≤ b ∗ /γ on [0 , s ( t )] for t > .Now, we show Theorem 6.2 by contradiction. Let us assume that there exists a constant C > such that s ( t ) ≤ C for t ∈ [0 , T ] . Then, by Lemma 6.3, we have that (cid:90) t | s t ( τ ) | dτ + (cid:90) t | u z ( τ ) | L (0 ,s ( τ )) dτ + (cid:90) t (cid:12)(cid:12)(cid:12)(cid:12) u ( τ, − b ∞ γ (cid:12)(cid:12)(cid:12)(cid:12) dτ ≤ C for t > , (6.12) (cid:90) t | u t ( τ ) | L (0 ,s ( τ )) dτ + | u z ( t ) | L (0 ,s ( t )) ≤ C for t > . (6.13)Here, by (1.4), we see that s t ( t ) ≥ , and hence s ( t ) ≥ s for t > . Also, u ≤ b ∗ /γ on [0 , s ( t )] for t > so that it holds that | s t ( t ) | ≤ a b ∗ /γ for t > . From these results and the change of variables(2.1) we obtain that (cid:90) t | ˜ u t ( τ ) | L (0 , dτ = (cid:90) t (cid:90) s ( τ )0 s ( τ ) | u t ( τ, z ) + u z ( τ, z ) zs ( τ ) s t ( τ ) | dzdτ ≤ (cid:90) t (cid:90) s ( τ )0 s ( τ ) | u t ( t, z ) | dzdτ + (cid:90) t (cid:90) s ( τ )0 s ( τ ) | u t ( τ, z ) || u z ( τ, z ) || s t ( τ ) | dzdτ + (cid:90) t (cid:90) s ( τ )0 s ( τ ) | u z ( τ, z ) | | s t ( τ ) | dzdτ ≤ s (cid:18) C + 2 a b ∗ γ C / C / + (cid:18) a b ∗ γ (cid:19) C (cid:19) for t > , (6.14)and | ˜ u y ( t ) | L (0 , = (cid:90) s ( t )0 s ( t ) | u z ( t ) s ( t ) | dzdt ≤ CC for t > , (6.15)where C and C are positive constants as in (6.12) and (6.13).Here, for { t n } such that t n → ∞ as n → ∞ , we put ˜ u n ( t, y ) := ˜ u ( t + t n , y ) for ( t, y ) ∈ [0 , × [0 , , s n ( t ) := s ( t + t n ) , b n ( t ) := b ( t + t n ) for t ∈ [0 , . By (A2)’, it is clear that b n → b ∞ in L (0 , as n → ∞ . Also, by (6.12), (6.14) and (6.15) we see that { s nt } is bounded in L (0 , and { ˜ u n } is bounded in W , (0 , L (0 , ∩ L ∞ (0 , H (0 , . Therefore, we can take a subsequence { n j } ⊂ { n } such that the following convergences holds for some ˜ u ∞ ∈ H (0 , and s ∞ ∈ R satisfying s ≤ s ∞ < + ∞ : ˜ u nj (0) = ˜ u ( t nj ) → ˜ u ∞ in C ([0 , , weakly in H (0 , ,s nj (0) = s ( t nj ) → s ∞ in R , ˜ u njt → in L (0 , L (0 , and s njt → in L (0 , , ˜ u nj → ˜ u ∞ in C ([0 , L (0 , , weakly in W , (0 , L (0 , , weakly -* in L ∞ (0 , H (0 , ,s nj → s ∞ in C ([0 , , weakly in W , (0 , free boundary problem describing migration into rubbers – quest of the large time behavior j → ∞ . Also, by Sobolev’s embedding theorem in one dimension (3.17), it holds that for y ∈ [0 , , (cid:90) | ˜ u nj ( t, y ) − ˜ u ∞ ( y ) | dt ≤ C e (cid:90) | ˜ u nj ( t ) − ˜ u ∞ | H (0 , | ˜ u nj ( t ) − ˜ u ∞ | L (0 , dt. (6.16)Hence, by the strong convergence of ˜ u nj and (6.16) we see that ˜ u nj ( y ) → ˜ u ∞ ( y ) in L (0 , at y = 0 , as j → ∞ . (6.17)Now, for each j , ( s nj , ˜ u nj ) satisfies ˜ u njt ( t, y ) − s nj ( t ) ˜ u njyy ( t, y ) = ys njt ( t ) s nj ( t ) ˜ u njy ( t, y ) for ( t, y ) ∈ Q (1) , − s nj ( t ) ˜ u njy ( t,
0) = β ( b nj ( t ) − γ ˜ u nj ( t, for t ∈ (0 , , − s nj ( t ) ˜ u njy ( t,
1) = ˜ u nj ( t, s njt ( t ) for t ∈ (0 , ,s njt ( t ) = a ˜ u nj ( t, for t ∈ (0 , . By letting j → ∞ in the above system and using the strong convergences of ˜ u nj and s nj , we seethat ˜ u ∞ ∈ H (0 , and − s ∞ ˜ u ∞ yy = 0 on (0 , . (6.18)Hence, by using the above convergences of ˜ u nj and s nj , (6.17) and (6.18) we infer that ( s ∞ , ˜ u ∞ ) satisfies − s ∞ ˜ u ∞ y (0) = β ( b ∞ − γ ˜ u ∞ (0)) , − s ∞ ˜ u ∞ y (1) = 0 , ˜ u ∞ (1) = 0 . Therefore, we see that ( s ∞ , ˜ u ∞ ) is a solution of (˜ P ) ∞ such that ˜ u ∞ ∈ H (0 , and s ≤ s ∞ < + ∞ .This contradicts that (˜ P ) ∞ does not have a solution (see Lemma 6.1). Thus, we conclude that s goesto ∞ as t → ∞ and Theorem 6.2 holds. In this section, we use our free boundary model to approximate numerically the diffusion of apopulation of solvent molecules (cyclohexane) into a piece of material made of ethylene propylenediene monomer rubber (EPDM). The actual migration experiment and the set of basic parametersare reported in [17].In this framework, we take the effective diffusivity with an order of magnitude higher and explorebriefly of the depth of the penetration front depending on variations in the kinetic parameter a arising in (1.4). In fact, we look only at a particular instance of the large-time behavior of ourproblem and point out that, depending on the choice of model parameters, the free boundary position0 Kota Kumazaki, Toyohiko Aiki and Adrian Muntean D i ff u s i o n f r o n t , [ mm ] Numerical resultExperiment data (a) Comparison to experimental data.
Log(time) L o g ( d i ff u s i o n f r o n t ) t -behavior a =0.2 a =2 a =4experimental points (b) Penetration fronts for various values a . Figure 1: Approximation of the large-time behavior of solutions to ( P )( u , s , b ) on [0 , T ] with T = 5000 minutes. s ( t ) behaves like a power law of type t β , where β is typically different than or as expected forthe classical diffusion and for the Case II diffusion, respectively; see [12] for a detailed discussionbased on first principles on the large time behavior of sharp diffusion fronts in the transition fromglassy to rubbery polymers.As shown in Figure 1 (a), the behavior of our free boundary seems to be different from the realexperimental result. From a phenomenological point of view, a more realistic behavior of the freeboundary is obtained in [17]. On the other hand, Theorem 6.2 guarantees that the growth observed innumerical results is correct, theoretically. Moreover, in order to measure the growth rate for the freeboundary, we show numerical results for varying positive constants a in Figure 1 (b). From theseresults we conjecture that the free boundary position corresponding to Figure 1 (a) behaves like t . .This is a sub-diffusive regime. However, other parameters can bring the front in a super-diffusiveregime. Based on our current simulation and mathematical analysis results, we can only state thatwe expect the free boundary position to follow a power law for large times, but we are, for themoment, unable to establish rigorously quantitative upper and lower bounds on s ( t ) . Nevertheless,relying also on results from [10], we hope to be able to adapt some parts of our working techniquedeveloped in [3] to handle this case. The main difficulty lies on the fact that it seems that, for alarge region in the parameter spaces, our sharp diffusion fronts tend to deviate from t . This makesus wonder what is the most relevant exponent β and also for which parameter case and type(s) ofrubber-like materials this corresponds. We were able to prove the global solvability for a one-phase free boundary problem with nonlinearkinetic condition that is meant to describe the migration of diffusants into rubber. Despite of itsapparently simple one-dimensional structure, our free boundary model brings in a number of openquestions. The most important ones include the identification of an asymptotic dependence of type free boundary problem describing migration into rubbers – quest of the large time behavior s ( t ) ∼ O ( t β ) as t → + ∞ and its rigorous mathematical justification. Also, capturing numericallythe large time behavior so that a certain power law is preserved requires a special care; compare e.g.the ideas from [6, 19] to be adapted for the finite element method used here; see [17] for a detaileddescription of the numerical scheme used in this context. Of course, to bring the one-dimensionalmodel equations to describe better the physical scenario of diffusants migrating into rubbers, moremodeling components must be added, viz. macroscopic swelling, capillarity transport. The case ofmore space dimensions is out of reach as it is not at all clear how the kinetic condition on the movingsharp diffusion front should be formulated especially close to corners or other singularities of thegeometry. Acknowledgments
T. A. and A. M. thank the KK Foundation for financial support (project nr. 2019-0213). The workof T. A. is partially supported also by JSPS KAKENHI Grant Number JP19K03572, while theone by K.K. is partially supported by JSPS KAKENHI Grant Numbers JP16K17636, JP19K03572and JP20K03704. Fruitful discussions with U. Giese, N. Kr ¨oger, R. Meyer (Deutsches Institut f ¨urKautschuktechnologie, Hannover, Germany) and S. Nepal, Y. Wondmagegne (Karlstad, Sweden)concerning the potential applicability of this research in the case of diffusants migration into poly-mers have greatly influenced our work.
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