Existence of periodic solution of one dimensional free boundary problem for adsorption phenomena Текст научной статьи по специальности «Математика»

Онлайн-доступ к журналу: http: / / mathizv.isu.ru

Серия «Математика»

2018. Т. 25. С. 3-18

УДК 518.517

MSG 35R35, 35К60, 35В40

DOI https://doi.Org/10.26516/1997-7670.2018.25.3

Existence of periodic solution to one dimensional free boundary problems for adsorption phenomena

Japan Women's University, Tokio, Japan

N. Sato

National Institute of Technology, Nagaoka College, Niigata, Japan

Abstract. In this paper we consider a drying and wetting process in porous medium to create a mathematical model for concrete carbonation. The process is assumed to be characterized by the growth ol the air zone and a diffusion ol moisture in the air zone. Under the assumption we proposed a one-dimensional free boundary problem describing adsorption phenomena in a porous medium. The free boundary problem it to find a curve representing the air zone and the relative humidity of the air zone. For the problem we also established existence, uniqueness and a large time behavior of solutions. Here, by improving the method for uniform estimates we can show the existence of a periodic solution of the problem. Also, the extension method is applied in the proof. This idea is quite important and new since the value of the humidity on the free boundary is unknown.

Keywords: free boundary problem, periodic solution, fixed point argument.

Recently, we have proposed and studied the following free boundary problem (1.1) - (1.5) to investigate concrete carbonation process. In this model we consider a drying and wetting processes in one hole of a porous medium and regard the hole as a one-dimensional interval [0,L], where L is the length of the hole. Also, we suppose that the interval consists of the water-drop (liquid) zone [0,s(i)) and the air zone (s(t),L], and denote by u the relative humidity in the air zone, where t € [0, T] is the time variable and x = s is a curve with 0 < s < 1 on [0, T] for T > 0 (see Figure 1).

T. Aiki

1. Introduction

The problem is to find a pair of the curve x = s(t) and the function u

on

QS(T) := {(i, x); 0 < t < T, s(i) < x < L} (see Figure 2) satisfying

Figure l.\ Image of one hole

Figure 2.\ Domain of the problem

РдЩ - KUXX = 0

u(t, L) = b(t) d

in QS(T), for 0 < t < T,

¿(t)(:=-S(i))=aWM(t))-^(t)))

for 0 < i < Г,

KUx(t, s(t)) = (pa - pgu(t, s(t)))s(t) for 0 < t < Г, s(0) = so, it (ж, 0) = ito(x) for so < ж < L,

(l.i) (1.2)

(1.3)

(1.4)

(1.5)

where pa and pg are constants of the density of the aqueous-H^O and the gaseous-H20, respectively, k is a diffusion constant of the gaseous-H^O, a positive constant a and a continuous function y? on R indicate the growth rate of the liquid zone, b is a given boundary function on [0, T], and so and uo are initial data.

This problem was first proposed in 2013 [5]. The main idea of the modeling is to regard that the degree of saturation is ratio of the length of the liquid region to the whole length. By this assumption we can describe the relationship between the degree of saturation and the relative humidity by the free boundary problem. Before this research we adopted a play operator as a mathematical description for the relationship, for example, [2; 3; 10]. However, the system with the play operator model has a difficulty on regularities of solutions. Then, we arrived at the free boundary model to overcome the difficulty.

In [5; 12] under suitable conditions for a, ip, 6, pa, pg, so and uq we have proved that the above initial-boundary value problem P(6, so, uq) := {(1.1) (1-5)} admits a unique solution {s,w} on [0, To] for some 0 < To < T. Also, we [6] showed the global existence of a solution in time and convergence of the solution to the stationary state as t —)> oo by getting uniform estimates of the solution with respect to t. Moreover, this free

boundary problem is a part of a two-scale model for concrete carbonation process in three dimensional domain Q c K3 (see [4]). On this subject Kumazaki [8;9] obtained continuity and measurability of solutions of P(6(£), so(£)> uo(0) with respect to £ € Q,, when the function &(£) = 6(t, £), so(£) and uo(0 = uo(x,£) are defined for t € (0,T),x € (so(£), L),£ £ Q. Furthermore, in our recent work [11] the local existence of a solution of the two-scale model was proved.

In this paper for 0 < T* < oo we consider a periodic problem PP(fe) := {(1.1)-(1.4), (1.6)},

s(0) =s(n),u(0) =u(n) on (s(0), L), (1.6)

and establish the existence of a periodic solution by improving a way to get uniform estimates. There are a lot of results dealing with periodic solutions of a classical one-dimensional Stefan problem which is one kind of free boundary problems. For example, in [1] existence and uniqueness of a periodic solution were proved. Particularly, in the proof of the uniqueness the weak formulation called enthalpy formulation plays a very important role (also see [7]).

However, in the present model a weak formulation is not found, yet. Thus, the uniqueness of the periodic solution of PP(6) is open problem, now.

We define a solution of our problem and give a statement of our main theorem in the next section. In section 3 we shall obtain some uniform estimates for solutions with respect to t. Finally, we prove the theorem in section 4.

2. Main result

We begin with assumption for given data if, a, pg, pa, L and b. (Al) <p € C^K) n t71,0°([R), <p = 0 on (-oo, 0], <p < 1 on R and f' > 0 on (0 ,L\.

(A2) a, k, pa and pg are positive constants satisfying

pa > 2pg,pa > pg ^(1 + |^|l°°(r) + 2^ and 9aLp2g < npa.

(A3) b G W1'2^,^), 6(0) = b(Tt) and 0 < b < on [0,T*], where is a positive constant with < <p(L). We put do = <f(L) — 6* > 0.

Here, we define solutions of P(6, so, uo) and PP(6) in the following way:

Definition 1. Lets and u be functions on [0, T] andQs(T), respectively, for T > 0. We call that a pair {s,u} is a solution of P(b,so,uo) on [0,T] if the conditions (SI), (S2) and (1.1) - (1.5) hold:

(51) s € Wl'°°(<d,T), 0 < s < L on [0,T],

(52) и € L°°(QS(T)), щ,ихх € L2(QS(T)), \ux(-)\L2{s{.)tL) € L°°(0,T). Also, it is called that the pair {s,-u} is a solution of PP(b) on [0, T*], if (SI) and (S2) hold with T = Г . and (1.1) - (1-4), (1-6) are valid.

Remark 1. Let {s,-u} be a solution of P(6, so, uo) on [0,T], T > 0. It is easy to see that и € C(Q(T)). Immediately by (1.3) we have s € C'1([0, T}).

The main result of this paper is as follows:

Theorem 1. If (Al) - (A3) hold, then the periodic problem PP(b) has a solution on [0,T*].

Before we give a sketch of a proof of Theorem 1, we recall existence and uniqueness results for P(b, so,uo).

Theorem 2. [6, Proposition 2.3, Theorem 2.4] If (Al) - (A3) and (A4) hold,

(A4) 0 < so < L, uo G H\s0, L),uo(L) = 6(0) and 0 < uo < 1 on [so,L],

then P(b, so, uo) has one and only one solution {s,u} on [0,T*] satisfying 0 < и < 1 on Qs(T

Next, in order to consider our problem on the cylindrical domain we introduce some notations. Let {s,u} be a solution of P(6, so, uo) on [0,T*] and put

ù(t,y) = u(t, (I-y)s(t)+Ly) for (t,y) € Q(T*) := (0 ,T,) x (0,1). (2.1) Then ù satisfies:

Ря** ~ (L-s)^yy = PgiL-fSjly in g(T*}' (2"2)

ù(t,l)=b(t) for 0 < i < T*, (2.3)

s(t) = a(v,(t, 0) - (p(s(t))) for 0 < t < Г», (2.4)

L^s^ùy(t, 0) = (pa - pgu(t, 0))s(t) for 0 < t < T*, (2.5)

s(0) = so,u(0, y) = ùo(y) := «o((l - v)so + Ly) for 0 < у < 1(2.6)

Here, we provide the sketch of the proof in which we apply the Schauder fixed point theorem.

1) For any M > 0 and ô > 0 we define a set

= |0,1 - i, X { ^^j'iTf^ Wi) } •

Известия Иркутского государственного университета. 2018. T. 25. Серия «Математика». С. 3-18

For (s0, u0) € K(M, 5) we write uo = mo|[so,l]> again, where mo|[so,l] is a restriction of -uo to [so, L]. Then, Theorem 2 implies the existence of a solution {s, u} of P(6, so, Mo) on [0, T*]. Hence, we can define a solution operator A : K(M,5) -»• [0,1) x L2(0,L) by A(s0,u0) = (s(T*), u(T*)), where -u(T*) is an extension of -u(T*) defined below:

,T v j u(T*,x) for s(T*) < x < L, .

UV*>X)- \ u(T*,s(T*)) forO <x<s(T*). {ZJ)

2) By using uniform estimates obtained in Lemmas 2 and 4 we can take M > 0 and 5 > 0 such that A : K(M, 5) ->■ K(M, 5).

3) On account of the continuity of A given in Lemma 5 we can apply the fixed point theorem to A so that we can prove Theorem 1.

3. Uniform estimates with respect to t

In this section we provide some uniform estimates for the solution of P(6, s0, u0).

Lemma 1. Assume (Al) - (A4)- Let {s,-u} be a solution of P(b, so, uq) on [0,2;]. Then it holds that

K\ux(t,s(t))\ <2a(pa + pg),\s(t)\ <2a forte[0,T*]. (3.1)

Proof Since 0 < u < 1 on QS(T*), by (1.3) and (1.4) we see that

|s(i)| = a\u(t, s(t)) - <p(s(t))\ < 2a,

and

k\Ux(t,s(t))\ = \pa- pgu(t,s(t))\\s(t)\

<2a{pa + pg) for t € [0,T*].

Thus we have proved this lemma. □

Now, we give essential uniform estimates in the proof of Theorem 1. Although the proof is quite similar to that of [6, Lemma 3.2], it is improved and the calculations are delicate so that we show it in detail, here. To state the lemma, we put

n fL

= = / \ux(t)\2dx for t € [0,2;].

2PgL Js(t)

Lemma 2. Under the same assumptions as in Lemma 1 let {s,-u} be a solution of P(b, so, uoj on [0,T*]. Then there exists a positive constant M\ depending only on pa, pb, <p, a, b and L (independent of sq and uq) such that

E(t) < e~KotE(0) + Mi for 0 < t < T*. (3.2)

Proof First, to handle the boundary condition, easily, we define a function tp on [R by

Уаг2 - \pg for r > 1, ф(г) = { \par2 - \рягъ for 0 < r < 1, (3.3)

0 for r < 0.

Clearly, tp is convex and continuous on [R. Moreover, we use the notation u and uo given by (2.1) and (2.6), respectively, and put Vh(t,y) = u(t, y)-u(t-h1yl{oTh>Q ^ ^ ^ where

h

= Tl<0n°-l-h (3.4)

v \ uo(0) for t < 0, у < 0, v '

and s(t) = s(0) + ts(0) and b(t) = 6(0) for t < 0. Here, we note that Remark 1 implies s € C1([0, T*]) so that the extension of s is available.

We multiply (2.2) by (L — s)vh and integrate it over (0,1) with respect to y. Then, we have

pg{L-s{t))l v,t{t)vh{t)dy Jo

ùy{t)vhy{t)dy + K MUy(t, 1 )vh(t, 1) (3.5)

L-s(t) J0 yv y yv y y L-s(t)

к ^

L - s(t) Vh<yt' ^ + Pgè® J0 ^ ~ y">ûy(t)vh(t)dy

4

(:=J^I~i(t)) for a.e. t € [0,T*]. i= 1

It is easy to see that

h(t) < ~,T f\\uy(t)\2 - \uy(t - h)\2)dy

Jo

2h (L - s(t)) JQ

~ ( [ |ux(t)\'^ .

2h \Js[t) L — s(t) Js[t_h)

^ -tS: f f \ux(t)\2dx- L r ^ [ \ux(t-h)\2dx

^ f I \ux(t)\2dx- f \ux(t-h)\2dx)

Ztl \Js(t) Js(t-h) J

Ks(t) -s(t-h) fL {Ux{t_h)l2dx for a.e. t € [0,T*],

2tl L - S{t) Js(t-h)

and

. . k , , bit) - bit - h)

^ = L^wfy^ h

= KU^L)*®'1^'® for a.e. t € [0,T*].

On account of (2.4) and (2.5) we infer that

h(t) = -a(pa-pgu(t,0))(u(t,0)-<p(s(t)))vh(t,0)

= -a(pa - pgu{t, 0))u(t, 0)vh{t, 0) + a{pa - pgu(t, 0))<p(s(t))vh(t, 0) := h,i(t) + J3>2(i) for a.e. t € [0,T*].

Here, since 0 < u < 1 on Q(T*), it holds that (pa — pgu(t,0))u(t,0) = ip'(u(t, 0)) for a.e. t € [0,T*], and from the convexity of tp it follows that

/3>i(i) < ~(ip(u(t,0))-tp(u(t-h,0))) for t € [0,T*].

Also, by using (2.4) again, we have

's(i) — s(t — h)

h,2(t) = (Pa~P9u(t,0)Ms(t))

h

+a(pa-pgü(t,0)Ms(t)) i^isit)) fs{t k)) ) for te[h,n\,

and

/3,2 (i)

= (Pa - pgü(m<p№) (m ~ f ~h)

+a(ptt - PgümMsit)) + for t e

Here, we integrate (3.5) over [0,ti] for 0 < t\ < T*. Accordingly, the above arguments lead to:

pg [ (L- s(t)) [ ùt(t)vh(t)dydt(:= îQh) < Jo Jo

ft! ( fL

~ 2hJ0 [Js(t)

ux(t)\2dx— / \ux(t — h)\2dx \ dt+ J s(t—h) J

к fbl s(t) - s(t - h) fL . , ,,,2, ,

fti ШЛ-Ш-^Л fti a

+ / M^(i,L) ——£--dt- / -(ip(ù(t,0))-ip(ù(t - h,0)))dt+

Jo "> Jo I1

+pa J^ ф{1)уу> ^-V-dt+apaJ ^(gffl) д -~dt-

-if J s(tMs(t))-±!—±--dt — pg J^ visit))2 v ; ^-!-dt-

-pgj^m^(t)fm)~\{s{t~h))dt-

Jo ">

fh, , ^^ , , ^ f Ф(0) + (t ~ h)s(0)) - <p(s(0))\

+ a / ipa — pgu(t, 0))ip(s(t)) 1 y J V ' y " >> \

Jo

h

+

+ pg [ sit) [ (1 - y)v,y(t)vh(t)dydt Jo Jo

(■■= H for апУ tl G [°>T*]- (3-6)

i= 1

First, by change of the variable we see that

lim/0h = P9 ( (L-s(t)) [ \ut(t)\2dydt = ^o J о y0

= pg / |-ut(t)|2cfedt+

Jo Js(t)

fti fL

f1 f L — x

+ 2pg / s(i) / --7-rut(t)ux(t)dxdt+

Jo Js(t) L ~ SW

Next, we have

K

fil f-L f 0 f-L

hh = -^r\ I I Iux(t)\2dxdt- / \ux(t)\2dxdt ¿I1 yJti—h Js(t) J—h Js(t) j

As mentioned in the proof of Lemma 3.5 in [12], the function

r-L

|2,

t —> / Mi) I dx Js(t)

is absolutely continuous on [0,T*]. Then it follows that

lim — / / \ux(t)\2 dxdt =— — / \ux(ti)\2dx.

h^o 2nJtl_hJs(t) ?< Js(ti)

The extension (3.4) leads to

, , _ ( uo(x) for t < 0, so < x < L, i(t,x) — < „, / \ w , ^ n ^ ^

U{t,x) I Uo(so) for t < 0, X < So-Then, it is easy to see that

k f° fL k fL 1 f° r(0)

—r / \ux(t)\2dxdt = - \uox\2dx + - / / I«oxI

ltlJ-hJs{t) 1 Js0 11 J-h Js0+ts(0)

K fL

—> — / \v,ox\2dx as h —> 0. 2 J so

Thus, we get

rL ,„ rL

¿""-'ira 0 / I^x-viyi I 0

2 Js(ii) 2 ^so

lim[ Iih = -- I \ux(ti)\ dx + - I \u0x\ dx.

Similarly,

, f K f S(t) f I / M2 , ,

hm l2h = — / t-7TT / dxdt.

h^O 2 Jo ^ - «CO

By (1.1) and (1.4) it follows that

lim I3h = f nux(t, L)b(t)dt = h^r о J о

= k[ [( uxx(t)dx + ux(t, s(t)) ) b(t)dt = J0 \Js(t) J

= [ \f pgUt(t)dx + (pa ~ pgf(s(t)))s(t)\b(t)dt <

Jo \Js{t) J

< тг Г Г \ut(t)\2dxdt + ^ Г \b(t)\2dt+

2 Jo Js{t) 2 jо

+ f\W)\2 + W)\2)dt.

Since u is continuous on Q(T*), we can get

lim Iau = — lim ( ~~ I ib(u(t,0))dt) + atb(uo(0)) h^o h^o \hjtl_h J

= -a(^(u(ii,s(ii))) -VCuo(O))). Here, by using s € C([0,T*]) and <p(s) € C1([0, T*]) we observe that

lim/ im f'-MsW)-^^))

/i^o /i^o 70 h

+ palimfi/1 f(s(t))s(t)dt-\ f <p(s(t + h))s(0)dt] = h^o\hJtl_h h J0 J

fti

= -Pa / </(s(i))|s(i)|2di + Po(^(s(ii))s(ii) - f£>(s(0))s(0)), Jo

\imi6h = ^(f(s(tl))2-f(sm2),

/i—s-0 Z

lim sup Ijh < lim sup [ ip(s(t))(s(t - h)2 - s(t)2)dt < h^o h^o 2an J о

Pa Lp(s(t + h)) - <p(s(t)) , , о 7

< lim sup Zf- -—v " s(t)2dt+

h^rо 2a y0 "

+ ^limsupfi [ f(s(t))s(t)2dt - \ [ f(s(t + h))s(0)2dt] <

2a \n Jtl-h iT-J-h J

< ^ J*1 <p'(s(t))\s(t)\3dt + ^ (¥>(*(*l))l*(tl)|2 - ^(S(0))|i(0)|2)

lim ith = lim f Mttf-Mt + h)?

h^r0 M h^r0 Jo h

- i™ {l r_h ^W)2^ - \ f_h ^ + h))2s(0)d?j =

= 2pg f1 v'(s(t)Msmm\2dt - pfl(^(s(ii))2s(i!) - t£>(s(0))2s(0)), J 0

lim igh = f1 \s(t)\2v(s(tW(s(t))dt < 0, o a Jo

jimho^-^sih))3-^))3),

/i—)• 0 O

and

If , / M /, fh ,

\hih\ < a(pa + P9W\l°°(r) Jo 1-h^ ^ as /w 0.

Moreover, we see that

f1 f L — x

lim I\2h = Pg s(t) / 7-7-ux{t)ut{t)dxdt+

Jo Js{t) L - s\t)

+p°Cmi2L {r^y M)i2dxdt

ftl . fL / T — rr \ 2

s(t)

From the estimates for I0h, • • • , I\2h it follows

7T I' f \ut(t)\2dxdt + ^- [ |Uxih^dx + Fih) < 1 Jo Js(t) 1 Js(t!)

ft i fL L — x

< F(0) - pg / s(t) / --j-ut{t)ux{t)dxdt+

Jo Js(t) L ~ SW

+ J [ \u0x\2dx+ [ ( [ \ux(t)\2dxdt+

2 Jso Jo \2{L ~ S{t))J Js{t)

fti

+ Ci / (\b(t)\2 + 1 )dt for 0 < h < T*, (3.7) Jo

where C\ is a positive constant depending only on pa, pg, a, if, k, and L and

F(t) = a^u(t,sm-Pa^s(t))s(t)-^(s(t))2-^(smm\2 +^{s{t)f + pgV(s(t))2s(t) for 0 < t < T*.

O

Furthermore, for the second term in the right hand side of (3.7) we can get

f*i fL L — x

~ Pg / s(t) / T-j-rut{t)ux{t)dxdt =

Jo Js(t) L ~ SW

= -n J s(t) J ^ ^uxx(t)ux(t)dxdt =

fH s(t) rL lo (L - s(t)) Js{t)

/ Id

(L - x)- — \ux{t)\2dxdt =

= ~n ! оf/^lf+w l \ux(t)\2dxdt+

Jo AL - Js(t)

Г* i

+ [ 1 s(t)\ux(t, s(t))\2dt for 0 < h < T* 2 Jo

By substituting this equation and (1.1) into (3.7) we have

77" i f \uxx(t)\2dxdt + ^; [ \ux(ti)\2dx + F(ti) < ¿Pg Jo Js(t) 1 Js{t{)

\uOx\2dx + F(0) + C2 [ 1 (\b(t)\2 + l)dt fori! e [0,T*], (3.8)

1 J so JO

where C2 is a positive constant depending only on pa, pg, a, if, k and L. Here, it is easy to see that

pL rL

/ \ux{t)\2dx < 2L2 / \uxx(t)\2dx + 2L\ux(t, s(t))|2 for i € [0,2;].

(3.9)

On account of (3.8) and (3.9) we see that

k2 [ll fL K

-r-F2 / \Ux(t)\2dxdt + -E(t1)+F(t1) <

4PgL Jo Js(t) 2

< +F(0) +C3 Td^t)!2 + l)di for t\ € [0,T*],

^ Jo

where C3 is a positive constant.

By regarding to € [0, T*] as the initial time the above argument implies that

ко [ 1 E(t)dt + + F(ii) <

< -E(to) + F(io) + c3 / (|6(i)r + l)di for 0<to<h< T*.

2 J to

Then, the absolutely continuity of E on [0,2;] guarantees that

KoE(t) + ~E(f) + i) < C3(|6(t)|2 + 1) for a.e. t € [0,T*](3.10) 2 at at

We multiply (3.10) by eKoi and have

jt(eKot(E(t) + F(t))) < C3eKot(\b(t)\2 + l) + K,0eKotF(t) for a.e. t € [0,T*] so that there exists a positive constant M\ such that

E(t) < e~KotE(0) + Mi for t € [0, T*]. Thus we have obtained (3.2). □

Next, we show the uniform estimate for the free boundary s:

Lemma 3. (cf. [6, Lemma 3.3]) Let {s,-u} be a solution of P(b, so, uq) on [0,2;] and M* be a positive constant satisfying f^ \ux(t)\2dx < M* for 0 < t < 21*, and put

s*m = L-( ) .

\ 2(V M* + C-^v L) J Ifs{t) > s*(M*) for some t € [0,2;], then s(t) < 0.

Proof. Assume that s(t) > s*(M*) for some t (E [0,2;]. Then we observe that

s(t) = a(u(t,s(t))-V(s(t))) =

= a(u(t, s(t)) - 6(i) + 6(i) - + - + - <p(s(t))) <

< a ( \/L — s(t)( [ \ux{t)\2dx)1'2 - d0 + C^L - s(t))) <

V J

< a (y/L - s^M^y/W* + C^vT) - do) <

This is a conclusion of this lemma. □

Lemma 4. ("c/. Proposition 3.4]) Let {s,-u} be a solution of P(b, so, uo) on [0,2;]. Then it holds that

s(t) < max{so, s*(M*)} for any t € [0,2;]. (3.11)

Proof. First, we assume that so > s*(M*). Then, since s is continuous on [0,2;], one of the following cases must occur:

(case 1) s(t) > s*(M*) for t € [0,2;].

(case 2) there exists to € (0, T*] such that

s > s*(M*) on [0,T0] and s < s*(M*) on (t0,t0 + 5') for some 5' > 0.

In case 1, by Lemma 3 we have s < 0 on [0, T*] so that s(t) < s(0) for t € [0,2;]. Thus, we get (3.11). In case 2, we suppose that there exists ii > to such that s(t\) > s*(M*). Then we can take t2 € (to,t\) satisfying s(t2) > s*(M*) and ¿(¿2) > 0, since s € C1([0,T*]). This contradicts to Lemma 3. Hence, (3.11) holds.

If so < s*(M*), then we can obtain (3.11) in the similar way. This lemma is proved. □

4. Existence of periodic solution

The aim of this section is to prove Theorem 1. First, we recall the continuous dependence of data for a solution of P(6, so, uo).

Lemma 5. (cf. [6, Lemma 4-3]) Assume (Al) - (A3). For any M > 0 and 5 > 0 let (soi,uoi) £ K(M,6), i = 1,2. Here, Theorem 2 implies the existence of a solution {si,Ui} of P(b, soi, uoi) on [0,2;] for each i. Here, by using (2.1) we define Ui and uoi from Ui and uoi, respectively, for each i. Then there exists a positive constant M2 such that

\si(t) - s2(t)|2 + |ui(i) - Mt)\2H + J \uiy ~ u2y\2L2{o,i)dT < M2 (|soi - S02I2 + \U01 - M02|l2(0>i)) for0<t< T*.

Proof of Theorem 1. Let M > 0 such that

Mi < (1 -e~KoT*)M,

where M\ is the positive constant obtained in Lemma 2. Moreover, we can take 5 > 0 such that s*(M) <1 — 5, where s*(M) is defined in Lemma 3. Then, Theorem 2 implies that for (so,uo) £ K(M,S) P(6, so, uq) has a unique solution {s,-u} on [0,2;]. Here, we extend «(T*) to the function on [0, L] by (2.7). Thanks to Lemma 2 we have

pL rL

/ \ux(T,)\2dx <e~KoT* / {uoxfdx + M^ Jo Jo

Easily, we see that J0L ¡UxiT^^dx < M. From Lemma 4 it follows that s(T) < 5. These facts indicate A : K(M, 5) ->■ K(M, 5).

Clearly, Lemma 5 leads to the continuity of A in [RxL2(0, L) and K(M, 5) is convex and compact in the topology of R x L2(0, L). Hence, the theorem is a direct consequence of the Schauder fixed point theorem. □

PERIODIC SOLUTIONS OF FREE BOUNDARY PROBLEM References

1. Aiki T. Periodic stability of solutions to two-phase Stefan problems with nonlinear boundary condition. Nonlinear Anal. TMA., 1994, vol. 22, pp. 1445-1474.

2. Aiki T., Kumazaki K. Mathematical model for hysteresis phenomenon in moisture transport of concrete carbonation process. Physica B, 2012, vol. 407, pp. 14241426.

3. Aiki T., Kumazaki K. Well-posedness of a mathematical model for moisture transport appearing in concrete carbonation process. Adv. Math. Sci. Appl., 2011, vol. 21, pp. 361-381.

4. Aiki T., Kumazaki K., Murase Y., Sato N. A two-scale model for concrete carbonation process in a three dimensional domain. Surikaisekikenkyusho Kokyuroku, 2016, no. 1997, pp. 133-139.

5. Aiki T., Murase Y., Sato N., Shirakawa K. A mathematical model for a hysteresis appearing in adsorption phenomena. Suurikaisekikenkyuusho Koukyuuroku, 2013, no. 1856, pp. 1-11.

6. Aiki T., Murase Y. On a large time behavior of a solution to a one-dimensional free boundary problem for adsorption phenomena. J. Math. Anal. Appl., 2017, vol. 445, pp. 837-854.

7. Damlamian A., Kenmochi N. Periodicity and almost periodicity of solutions to a multi-phase Stefan Problem in several space variables, Nonlinear Anal. TMA., 1988, vol. 12, pp. 921-934.

8. Kumazaki K. Continuous dependence of a solution of a free boundary problem describing adsorption phenomenon for a given data. Adv. Math. Sci. Appl., 2016, vol. 25, pp. 289-305.

9. Kumazaki K. Measurabllity of a solution of a free boundary problem describing adsorption phenomenon. Adv. Math. Sci. Appl., 2017, vol. 26, pp. 19-27.

10. Kumazaki K., Aiki T. Uniqueness of a solution for some parabolic type equation with hysteresis in three dimensions. Networks and Heterogeneous Media, 2014, vol. 9, pp. 683-707.

11. Kumazaki K., Aiki T., Sato N., Murase Y. Multiscale model for moisture transport with adsorption phenomenon in concrete materials. Appl. Anal., 2018, vol. 97, pp. 41-54.

12. Sato N., Aiki T., Murase Y., Shirakawa K. A one dimensional free boundary problem for adsorption phenomena. Netw. Heterog. Media, 2014, vol. 9, pp. 655-668.

Toyohiko Aiki, Doctor of Science (Mathematics), Professor, Department of Mathematics, Faculty of Science, Japan Women's University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo, 112-8681, Japan (e-mail: aikit@fc.jwu.ac.jp)

Naoki Sato, Doctor of Science (Mathematics), Associate Professor, Division of General Education, National Institute of Technology, Nagaoka College, 888, Nishikatakai, Nagaoka, Niigata, 940-8532, Japan (e-mail: naoki@nagaoka-ct.ac.jp)

Received 04-06.18

Существование периодических решений в одномерной задаче со свободной границей, описывающей адсорбционные явления

Т. Аики

Японский женский университет, Токио, Япония

Н. Сато

Национальный технологический институт, Ниигата, Япония

Аннотация. Рассматривается процесс сушки и смачивания в пористой среде для создания математической модели карбонизации цемента. Предполагается, что данный процесс характеризуется ростом воздушной зоны и диффузией влаги в воздушной зоне. При данном предположении предлагается одномерная задача со свободной границей, описывающая адсорбционные явления в пористой среде. Задача со свободной границей заключается в нахождении кривой, представляющей воздушную зону и относительную влажность воздушной зоны. Также устанавливаются существование, единственность и поведение решений на бесконечности. Также, улучшая метод равномерных оценок, показывается существование периодического решения задачи. Кроме этого, в доказательстве применяется метод расширения. Эта идея является весьма важной и новой, поскольку значение уровня влажности на свободной границе неизвестно.

Ключевые слова: задача со свободной границей, периодические решения.

Тойохико Аики, доктор физико-математических наук, профессор, математический факультет, Японский женский университет, Япония, 112-8681, Токио, Бунке, Медзиродай, 2-8-1 (e-mail: aikit@fc.jwu.ac.jp)

Наоки Сато, доктор физико-математических наук, доцент, колледж Нагаока, Национальный технологический институт, Япония, 9408532, Ниигата, Нагаока, Нисикатакай, 888, (e-mail: naoki@nagaoka-ct.ac.jp)

Поступила в редакцию 04-06.18