EXPONENTIAL STABILITY FOR A SWELLING POROUS-HEAT SYSTEM WITH THERMODIFFUSION EFFECTS AND DELAY Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 2, P. 65-77

YAK 517.955.8

DOI 10.46698/ y2253-0872-2762-l

EXPONENTIAL STABILITY FOR A SWELLING POROUS-HEAT SYSTEM WITH THERMODIFFUSION EFFECTS AND DELAY

M. Douib1 and S. Zitouni2

1 Department of Mathematics, Higher College of Teachers (ENS) of Laghouat, P. O. Box 4033, Laghouat 03000, Algeria; 2 Department of Mathematics and Informatics, University of Souk Ahras, P. O. Box 1553, Souk Ahras 41000, Algeria E-mail: [email protected], [email protected]

Abstract. In the present work, we consider a one-dimensional swelling porous-heat system with single time-delay in a bounded domain under Dirichlet-Neumann boundary conditions subject to thermodiffusion effects and frictional damping to control the delay term. The coupling gives new contributions to the theory associated with asymptotic behaviors of swelling porous-heat. At first, we state and prove the well-posedness of the solution of the system by the semigroup approach using Lumer-Philips theorem under suitable assumption on the weight of the delay. Then, we show that the considered dissipation in which we depended on are strong enough to guarantee an exponential decay result by using the energy method that consists to construct an appropriate Lyapunov functional based on the multiplier technique, this result is obtained without the equal-speed requirement. Our result is new and an extension of many other works in this area.

Keywords: swelling porous, well-posedness, thermodiffusion effects, delay term, exponential stability. AMS Subject Classification: 93D20, 35B40, 35L90, 45K05.

For citation: Douib, M. and Zitouni, S. Exponential Stability for a Swelling Porous-Heat System with Thermodiffusion Effects and Delay, Vladikavkaz Math. J., 2023, vol. 25, no. 2, pp. 65-77. DOI: 10.46698/y2253-0872-2762-l.

1. Introduction

In this paper, we study well-posedness and exponential stability for a swelling porous-heat system with thermodiffusion effects and delay. The system is written as /

piutt - aiuxx - d2fxx = 0,

P2ftt - a3^xx - a2Uxx - Yi9x - Y2PX + ft + №ft(x, t - T) = 0, cdt + dPt - fc^xx - Yifxt = 0, ^ddt + rPt - hPxx - Y2fxt = 0,

where (x,t) € (0,1) x (0, and we impose the following initial and boundary conditions

(1)

u(x,0) = uo(x), ut(x, 0) = u1(x), x € (0,1),

f(x,0) = : fo(x), ft(x, 0) = = f1(x), x € (0,1),

9(x, 0) = 9o(x), P(x, 0) = Po(x), x € (0,1),

ft(x, -t) = fo(x,t), t € (0, t),

u(0,t) = f (0, t) = 9(0, t) = P(0, t) = 0 (Vt ^ 0),

Ux(1,t) = = fx(1,t) = 9(1, t) = P (1,t) = 0 (Vt ^ 0),

© 2023 Douib, M. and Zitouni, S.

where u = u(x, t) is the displacement of the fluid and <p = <p(x, t) is the elastic solid material; p1 and p2 are the densities of u and <p, respectively; 9 = 9(x, t) is the temperature difference and P = P(x, t) is the chemical potential; k and h are heat and mass diffusion conductivity coefficients, respectively. The coefficients a1, a3 are positive constants and a2 = 0 is a real number such that a1a3 > ai,. The coefficients is positive constant and is a real number. Here, we prove the well-posedness and stability results for the problem (1)-(2), under the assumption

w > M- (3)

The physical positive constants y1, y2, r, c and d satisfying

A = rc - d2 > 0. (4)

Equations (1)1)2 are results of the basic field equations for the theory of swelling of one-dimensional porous elastic soils, given by (see [1])

ip1U« = T1x - P1 + F1, \p2Vtt = Tix - P2 + F2,

(5)

where Tj are the partial tensions, Fi are the external forces, and Pi are internal body forces associated with the dependent variables u and respectively. We assume that the constitutive equations of partial tensions as follows

Ti\ = iai a2\ fux\ (6)

T2J Vfl2

M

where M is a positive definite symmetric array, i. e., a2 < a1a3, and the internal forces of the body are considered null, that is, P1 = P2 = 0. We finally chose

Fi = 0 and F2 = ji6x + Y2Px - Vi<Pt - (x, t - t).

Time delay equations have a wide range of applications in the biological, mechanical social sciences, and many other modelling of the phenomena. It depend not only on the present state but also on some past occurrences. We know the dynamic systems with delay terms have become a major research subject in differential equation since the 1970s of the last century (e. g. [2-8]). It was shown that delay is a source of instability unless additional conditions or control terms are used (see [9]). On the other hand, it may not only destabilize a system which is asymptotically stable in the absence of delay, but it may also lead to will posedness (see [10, 11] and the references therein). Therefore, the stability issue of systems with delay plays great importance theoretical and practical in most of researches. In [8], the authors considered (5) by taking

Pi = P2 = 0, Fi = -^i^t - №<Pt(x, t - t) and F2 = 0,

they proved that the energy associated with the system is dissipative, and established the exponential stability of the system. Readers can consult [12-20] and the references therein for some other crucial results on the swelling porous system.

The purpose of this work is to study system (1)-(2), in introducing the delay term and thermodiffusion effects can make the problem different and crucial among the literature considered. The main features of this paper are summarized as follows. In Section 2, we

adopt the semigroup method and Lumer-Philips theorem to obtain the well-posedness of system (1)-(2). In Section 3, we use the perturbed energy method and construct Lyapunov functional to prove the exponential stability of system (1)-(2).

2. Well-Posedness

In this section, we prove the existence and uniqueness of solutions for (1)-(2). we introduce the new variable

z(x, p, t) = ipt(x, t - Tp), x € (0,1), p € (0,1), t> 0.

Therefore, problem (1) takes the form

p1Utt - a1Uxx - a2fxx = 0,

p2ftt - a3<Pxx - a2 Uxx - h9x - Y2Px + №t + №(x, 1, t) = 0, < TZt(x, p, t) + zp(x, p, t) = 0, c9t + dPt - k9Xx - 71<Pxt = 0, d9t + rPt - hPxx - Y2^xt = 0,

with the following initial and boundary conditions

u(x, 0) = uo(x), ut(x, 0) = u1(x), <p(x, 0) = po(x), Pt(x, 0) = (x), 9(x, 0) = 9o(x), P(x, 0) = Po(x), z(x,p, 0) = fo(x, Tp), z(x, 0, t) = <pt(x, t),

u(0, t) = p(0,t) = 9(0, t) = P(0,t) = 0

As in [7],

(7)

(8)

x € (0,1),

x € (0,1), x € (0,1),

(x,p) € (0,1) x (0,1), (x,t) € (0,1) x (0, +x>), (Vt ^ 0),

(9)

kUx(1,t) = ^x(1,t) = 9(1, t) = P(1,t) = 0 (Vt ^ 0). Introducing the vector function U = (u, ut, z, <p, <pt, 9, P)T. Then system (8)-(9) can be written

as

where the operator

/u\

Ut

V Vt z

e

\PJ

\U'(t) = A U (t), t > 0, \U (0) = Uo = (uo,u1,po ,P1,fo,9o,Po)T,

is defined by (

pi

[(Mifxx + a2uxx + Yiex + ~/2Px ~ vm - H2z{x, 1, t)}

(10)

ut

cx +

Vt

4- [ölUxx + 0>2<Pxx\

~rzp

'rk\ n _ (hd

\ x J xx VA

) Pxx + (a^l) Vtx

V

V ) Pxx ( A ) ®xx ( ^ A ^ I Vtx

/

Now, the energy space is defined by H = H1(0,1) x L2(0,1) x H1(0,1) x L2(0,1) x L2 ((0,1), L2(0,1)) x L2(0,1) x L2(0,1),

where

H1(0,1) = {/€ H1 (0,1); / (0) = 0}.

Let

U = (u,Ut,Lp,Lpt,Z,6,P)T, U = (u,ut,(p,(£>t,z,d,p) . Then, for a positive constant £ satisfying

TM < £ < T(2^1 — |№|), (11)

we define the inner product in H as follows 1

(U, U)M, = j [piutut + aiuxux + p2<-pt<f>t + a,3<-PxVx + «2 (uxtpx + ipxux)]dx 0

1 1 1

+ J [cM + d (P9 + 6P) + rPP ] dx + £j jz (x,p) z(x,p) dpdx. 0 0 0

The domain of A is

D(A) = {U € H | u,<p € H2(0,1), ut,<pt € H1(0,1),

e,P € H1(0,1), z, Zp € L2((0,1),L2(0,1))},

where

H2(0,1) = {/ € H2(0,1); /(0) = f*(1)=0}.

Clearly, D(A) is dense in H.

We have the following existence and uniqueness result.

Theorem 1. Assume that U0 € H and (4) holds, then problem (7)-(8) exists a unique solution U € C(R+; H). Moreover, if U0 € D(A), then

U € C (R+; D(A)) n C1 (R+; H) .

< To obtain the above result, we need to prove that A : D (A) ^ H is a maximal monotone operator. For this purpose, we need the following two steps: A is dissipative and Id — A is surjective.

Step 1. A is dissipative.

For any U = (u,ut,<p,<pt,z,e,P)T € D(A), by using the inner product and integrating by parts, we obtain

1

?2xdx-h I P2xa- 1 " ^ t

(At/, U)jf = —k J 9xdx — h J P2 dx — ^/xi — -^-^J J tp2 dx

0 0 0

1 1

— jj.2 J z(x,l,t) (ptdx —J z2(x,l,t)dx.

0

Using Young's inequality, we obtain

1 1 1

-H2 J z{x, 1, t) <pt dx < ^ J z2{x, 1 ,t)dx + ^ J <p% dx.

00

Therefore, from the assumption (11), we have

i

(AU, U)h < -k J eX dx — ^i

2t

2 J

VX dx

— h Px dx

2t

2 J

i

Iz2 (x' M) dx <0

Consequently, A is a dissipative operator. Step 2. Id - A is surjective.

To prove that the operator Id - A is surjective, that is, for any F = (fi,..., f7)T € H, there exists U = (u, ut, <p, <pt, z, 9, P)T € D(A) satisfying

(Id — A )U = F,

(12)

which is equivalent to

u — Ut = fi,

piUt — aiUxx — axVxx = pifx,

V — Vt = f3,

PxVt — a3Vxx — axUxx — Y^x — YxPx + ßiVt + ß2z(x, 1, t) = pxf4, (13)

TZ + Zp = Tf5,

\e — rkexx + hdPxx — (rYi — dY2) Vtx = Xf6, XP — chPxx + kdexx — (cY2 — dYi) Vtx = Xfj.

Suppose that we have found u and <p with the appropriate regularity. Therefore, the first and the third equations in (13) give

Ut = u — fi, Vt = V — fa-

(14)

It is clear that ut € H i(0,1) and tpt € Hi(0,L).

We note that the fifth equation in (13) with z(x, 0, t) = <pt(x,t), has a unique solution

p

z(x, p, t) = V(x)e-Tp — f3(x)e-Tp + Te-Tp J ersf5(x, s) ds,

(15)

clearly, z, zp € L2((0,1) x (0,1)).

By using (13), (14) and (15) the functions (u,<p,9,P) satisfy the following system

piU — aiUxx — axVxx = 9i, nV — aaVxx — axUxx — Y^x — YxPx = 92, xe — rkexx + hdPxx — (rYi — dYx)Vx = 93, XP — chPxx + kdexx — (cYx — dYi)Vx = 94,

i

i

where

n = P2 + + ^2e-r,

91 = P1f1 + P1 f2,

1

92 = P2f4 + nf3 — №Te-T J eTS f5(x, s) ds,

0

93 = a/G — (rY1 — dY2 )f3x, .94 = a/z — (cY2 — dY1) fax-

We multiply (16)i by u, (16)2 by (16)3 by f0, (16)4 by (16)3 by {P and (16)4 by {0 and integrate their sum over (0,1) to find the following variational formulation

B ((u, f, e, P)T, (u, f, 6, P)T) = G(u, f, 6, P)T, (17)

where B : [H1(0,1) x H1(0,1) x L2(0,1) x L2(0,1)]2 —► R is the bilinear form given by

1 1 1

B ^(u, f, e, P)T, (u, f, 9, = p1 J uu dx + a1 J uxux dx + a2 J (f xux + uxf x) dx

0 0 0 11 1111

+ nj ff dx + a3 J fxfx dx + cj ee dx + k J 6x6x dx + r J PP dx + hj PxPxdx

0 0 0 0 0 0 1 1 1

+ dj (6pP + P9) dx + Y1 J (efx — fx9) dx + Y2 J (Pfx — fxP) dx, 0 0 0

and G : [H (0,1) x H* (0,1) x L2 (0,1) x L2 (0,1)] —► R is the linear form defined by

11 11 1 1

c3{u,(p,9, P)T =Jgiudx+Jg20dx + ^ Jg30dx + ^ Jg<iPdx + — Jg3pdx + — Jg^Odx. 0 0 0 0 0 0

It is easy to verify that B is continuous and coercive, and G is continuous. So applying the Lax-Milgram theorem, we deduce that for all (u, f, (9, P) € H1(0,1) x H1(0,1) x L2(0,1) x L2(0,1), problem (17) admits a unique solution (u, f, e, P) € H1(0,1) x H1(0,1) x L2(0,1) x L2(0,1). The application of the classical regularity theory, it follows from (16) that (u, f, e, P) € H2(0,1) x H2(0,1) x hq(0,1) x hq (0,1). Hence, the operator Id — A is surjective. Consequently, the result of Theorem 1 follows from Lumer-Phillips theorem (see [21, 22]). >

3. Exponential Stability

In this section, we prove the exponential decay for problem (8)-(9). It will be achieved by using the energy method to produce a suitable Lyapunov functional. We define the energy functional E(t) as

1

1

E(t) = i J [pm2t + am2x + P2Vt + ¿Wx + 2a2ux<px + cd2 + 2ddP + rP2 ] da; o

1 1

^y J z2(x,p, t) dpdx.

1 1 (18)

I £ I [„2/ 2

0 0

Noting (4), we have for e,P = 0,

cd2 + 2ddP + rP2 = - e2 + (4= 0 + v/r^^ > o,

r

then we get that the energy E(t) is positive. The stability result reads as follows.

Theorem 2. Let (u,z,<p,9,P) be the solution of (8)-(9) and (4) holds. Then there exist two positive constants k0 and ki, such that

E(t) < koe-klt (Vt ^ 0). (19)

Before defining a Lyapunov functional, we need some lemmas as follows.

Lemma 1. Let (u,z,<p,9,P) be the solution of (7)-(8) and (4) holds. Then, the energy functional, defined by equation (18), satisfies

i i i i jt E(t) ^-k J 02x dx-h j P2 dx-C\ J (p2 dx-C2 j z2(x, 1, t) dx < 0, (20) o o o o

where

cx = - i^i - -L ^ 0, C2 = J--M>0.

p 2 2 T ' 2 r 2

< Multiplying (8)i, (8)3, (8)4 and (8)5 by ut, <pt, 9 and P, respectively, and integrating over (0,1) with respect to x, using integration by parts and the boundary conditions, we obtain

i

d

dt

J (piv" + aiv,x + pXVÎ + a3V2x + 2axUxVx + cex + 2deP + rPx) dx 0

i i i i

= —kj ex dx — hj P% dx — ßi J vX dx — ßX J Vtz(x, 1,t) dx.

On the other hand, multiplying (8)2 by |z(x,p,t) and integrating over (0,1) x (0,1), and recalling z(x, 0,t) = <pt, we obtain

i i i i

2ltJ j z2^x'P'^dPdx = ^ j'^t dx - ^ j z2(x,l,t)dx. (22)

o o o o

A combination of (21) and (22) gives

^ E(t) = -k J eX dx-h J P2 dx - (ßi - J Lp2t dx

0 0 0 i i

~ j z2(x,l,t) dx — ß2 j (fitz(x,l,t) dx.

1

Now, estimate the last term of the right-hand side of (23) as follows

1 1

-H2 J z{x, 1 ,t)<pt dx < ^ J z2{x, 1 ,t)dx + ^ J fl dx. (24)

0 0 0 Substituting (24) into (23), and using (11), we obtain (20), which completes the proof. > Lemma 2. Let (u, z,f, e, P) be the solution of (8)-(9). Then the functional

1

L1(t) = — p1 J uut dx, o

satisfies, for any e1 > 0, the estimate

1 2 1 1

L[(t) ^-pi J v2 dx + ^ai + ju2xdx + ei J Lp2xdx. (25)

o oo

< By differentiating L1(t) with respect to t, using (8)1 and integrating by parts, we obtain

1 1 1

L1(t) = — p1 J uf dx + a1 J ux dx + a2 J uxfx dx, ooo then, by Young's inequality, we obtain the result. >

Lemma 3. Let (u, z, f,e,P) be the solution of (8)-(9). Then the functional

1 1

L2(t) = a^y fft dx — a2P1 y fut dx,

o

satisfies, for any e2 > 0, the estimate

4C0 < J vldx + Cifa) J f2 dx + £2 J u2 dx + je„

0 0 0 0 1 1

+ Mi Jp2dx + Md j z\x^t)dx,

(26)

a x a

0 0

where 2

a = aia3-a2>0, C3(e2) = aip2 + ^^ +

a 4^2

< By differentiating L2(t) with respect to t, using the equations (8)1 and (8)2, and integrating by parts, we obtain

111 1

?x dx + a1P2 / f2 ^^ — I ^tu't /^1 I •yvx '

L2(t) = —^y fx dx + a1p^y f dx — a2p1 J ftut dx + Y1a^y fex dx

ooo o

1 1 1

+ Y2a1 y fPx dx — fft dx — ^2a1 J fz(x, 1,t) dx,

l

where a = a3ai - a^ > 0. Using Young's and Poincare inequalities, estimate (26) is established. >

Lemma 4. Let (u, z, 9, P) be the solution of (8)-(9) and (3) holds. Then the functional

i i r aip2 i , a3pi i ,

E3 (t) =- / LDtuax--/ utipax,

a2 a2

oo

satisfies, for any e3 > 0, the estimate

i i 2 i 2 i

L'3 (t) ^ -— u2xdx + a3 / ipl dx + 2fll271 / 92xdx + 2ai^2 / dx 2 J J a2 J a2 J

o o o o

i 2 i i + C2{e3) J Vt dx + J z2(x,l,t) dx + e3 J u2 dx,

(27)

oo

where

C4 (£3) - + —(aip2 ~ a3pl\ 2

43 a\ 4e3 \ a2 /

< By differentiating L3 (t) with respect to t, using the equations (8)i and (8)2, and integrating by parts, we obtain

i i i i 4(t) = -a, + ^dx + 3321 P.uir

o a2 a2

o o o

i i i

Viai i , V2ai f , . , , /aip2 - a3pi\ f

- / ifituax--/ z(x, 1, t)udx + - / LDtUtdx.

a2 a2 a2

o o o

Using Young's and Poincare inequalities, estimate (27) is established. >

Lemma 5. Let (u, z, 9, P) be the solution of (8)-(9) and (3) holds. Then the functional

i i

L4(t) = J J e-2Tpz2(x,p,t) dpdx, oo

satisfies, for some positive constants ni and n2, the estimate

i i i i L'4(t) ^ — n\ J J z2(x, p,t) dpdx — n2 J z2(x, 1, t) dx + — J ip2 dx. (28)

< By differentiating L4(t) with respect to t, and using the equation (7)3, we obtain

1 1

1 1

L'S) = 2 J Je 2Tt>z{xi Pit)zt{x, p,t) dpdx = —— J J e 2rpz(x, p,t)zp(x, p,t) dpdx

0 0 0 0 11 11

/ / J^(e~2rpz'2(x>P>t))dPdx ~2j j e~2rpz2(x,p,t) dpdx

00

11 1 1

^ — n\ J J z2(x,p,t) dpdx — n2 J z2(x,l,t) dx + — J f2 dx,

00

which gives the estimate (28). >

Now, we turn to prove our main result in this section.

< Proof of Theorem 2. We define the Lyapunov functional L(t) by

3

L(t) := NE(t) + ^ NiLi(t) + L4(t),

i=1

where N and Ni (i = 1,2,3) are positive constants that will be chosen later. By differentiating L(t), exploiting (20) and (25)-(28), we get

1

L'(t) < -[P1N1 - £2^2 - eaNa] y u2 dx -

Ux dx

CxN - C3(e2)N2 - C4(s3)N3 - -

T

f2 dx —

- N2 - a3N3 - eiNi

a

a:

dx

hN-^Nt-^Ns

a

a

0

f X dx

-■X d^x

1 1

- ni ¡J(x,p,t) ¿pi. -

00

a a22

z2(x, 1, t) dx.

At this point, taking ei = 1/Ni, i = 1,2,3. We then choose N1 large enough so that N1 > 2/p1. After that, we select N3 so that

Then, we choose N2 large enough so that

N2 - a3N3 - 1 > 0.

1

1

1

1

1

1

a

Finally, we select N large enough so that

CiN - C3(e2)N2 - C4(e3)N3 - - > 0,

T

a a22

a a22

C2N + n2-2-^N2-2-^N3>0. a a22

Consequently, from the above, we deduce that there exist a positive constant a0 such that

L'(t) < -aoE (t). (29)

On the hand, it is not hard to see that L(t) ~ E(t), i. e., there exist two positive constants ai and a2 such that

aiE(t) < L(t) < a2E(tt) (Vt ^ 0). (30)

Combining (29) and (30), we obtain that

L'(t) < -kiL(t) (Vt ^ 0), (31)

where k\ = A simple integration of (31) over (0,t) yields

L(t) < L(0)e-k11 (Vt ^ 0).

It gives the desired result of Theorem 2 when combined with the equivalence of L(t) and E(t). >

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Received January 12, 2022 Madani Douib

Department of Mathematics, Higher College of Teachers (ENS) of Laghouat,

P. O. Box 4033, Laghouat 03000, Algeria,

Professor of the Department of Mathematics

E-mail: madanidouib@gmail. com

https://orcid.org/0000-0002-5887-7652

Salah Zitouni

Department of Mathematics and Informatics, University of Souk Ahras,

P. O. Box 1553, Souk Ahras 41000, Algeria,

Professor of the Department of Mathematics and Informatics

E-mail: [email protected]

https://orcid.org/0000-0002-9949-7939

Владикавказский математический журнал 2023, Том 25, Выпуск 2, С. 65-77

ЭКСПОНЕНЦИАЛЬНАЯ УСТОЙЧИВОСТЬ ДЛЯ НАБУХАЮЩЕЙ ПОРИСТОЙ ТЕПЛОСИСТЕМЫ С ТЕРМОДИФФУЗИОННЫМИ ЭФФЕКТАМИ

И ЗАПАЗДЫВАНИЕМ

Дуиб М.1, Зитуни С.2

1 Кафедра математики Высшего педагогического колледжа Лагуата, Алжир, Лагуат 03000, а/я 4033;

2 Кафедра математики и информатики, Университет Сук-Ахрас, Алжир, 41000, Сук-Ахрас, а/я 1553 E-mail: [email protected], [email protected]

Аннотация. В настоящей работе рассматривается одномерная набухающая пористо-тепловая система в ограниченной области при граничных условиях Дирихле — Неймана с термодиффузионными эффектами и запаздыванием. Известно, что запаздывание без дополнительных предположений служит источником неустойчивости. Более того, введении запаздывания в асимтотически устойчивую систему может привести не только к потере устойчивости, но и к некорректно поставленной задаче. В этой связи исследование систем с запаздыванием на устойчивость имеет большое теоретическое и прикладное значение. Связанность системы вносит новый вклад в теорию, связанную с асимптотическим поведением набухания пористого тепла. Сначала мы формулируем и доказываем корректность решения системы полугрупповым подходом с использованием теоремы Люмера — Филипса при подходящем предположении о весе запаздывания. Затем получаем результат экспоненциального затухания, используя энергетический метод, основанный на методе умножения, в котором мы строим соответствующий функционал Ляпунова, этот результат получается без требования равной скорости. Наш результат является новым и является продолжением многих других работ в этой области.

Ключевые слова: набухание, пористость, термодиффузионные эффекты, запаздывающий член, экспоненциальная устойчивость.

AMS Subject Classification: 93D20, 35B40, 35L90, 45K05.

Образец цитирования: Douib M. and Zitouni S. Exponential Stability for a Swelling Porous-Heat System with Thermodiffusion Effects and Delay // Владикавк. мат. журн.—2023.—Т. 25, № 2.—C. 65-77 (in English). DOI: 10.46698/y2253-0872-2762-l.