Exponential stability for a swelling porous-heat system with thermodiffusion effects and delay

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

P i U tt - a i U xx - a 2 ^ xx = 0 ,

P 2 V tt — а з ^ хх — a 2 U xx — Y 1 9 x — Y 2 P x + Д 1 V t + №V t ( x, t — т ) = 0 , c9 t + dP t - k6 xx — Y i V xt = 0 , ^^d9 t ⁺ rP t ^— hP xx ^— Y 2 V xt = ⁰ ,

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

u(x, 0) = u o ( x ) , u t ( x, 0) = u 1 ( x ) ,	x € (0 , 1) ,
V(x, 0) = V o ( x ) , V t ( x, 0) = V i ( x ) ,	x € (0 , 1) ,
9(x, 0) = У о ( х ) , P ( x, 0) = P o ( x ) ,	x € (0 , 1) ,
V t ( x, - t ) = f o ( x,t ) ,	t € (0 , т ) ,
u (0 , t ) = v(0, t ) = 9(0, t) = P (0 ,t ) = 0	( V t ^ 0) ,
_u x (1 ,t ) = V x (1,t) = 9 (1 ,t ) = P (1 ,t ) = 0	( V t ^ 0) ,

where u = u ( x, t ) is the displacement of the fluid and y = y(x, t ) is the elastic solid material; p i and p 2 are the densities of u and y, 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 a i , а з are positive constants and a 2 = 0 is a real number such that a 1 a 3 > a ² ₂ . The coefficients µ 1 is positive constant and µ 2 is a real number. Here, we prove the well-posedness and stability results for the problem (1)–(2), under the assumption

№ >  l ^ 2 | .                                              ⁽³⁾

The physical positive constants γ 1 , γ 2 , r, c and d satisfying

A = rc — d ² >  0 .

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

f P i u tt = T i x - P i + F i , [P 2W = T 2 x - P 2 ⁺ F 2 ,

where T i are the partial tensions, F i are the external forces, and P i are internal body forces associated with the dependent variables u and ϕ, respectively. We assume that the constitutive equations of partial tensions as follows

T i ₌ a i a 2    u x

V T2)   Va 2 a 3 / XXM ’

M where M is a positive definite symmetric array, i. e., a22 < a1a3, and the internal forces of the body are considered null, that is, Pi = P2 = 0. We finally chose

F i =0 and F 2 = Y i 9 x + Y 2 P x - ^ i ^ t - ^ 2 ^ t ( x,t - т ) .

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 — ^2^t(x,t — т) 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). As in [7], we introduce the new variable

z ( x, p, t ) = ^ t ( x, t - тр ) ,    x E (0 , 1) , p E (0 , 1) , t >  0 .                   (7)

Therefore, problem (1) takes the form

P i U tt - a i u xx - a 2 ^ xx = 0 ,

P 2 V tt - а з ^ хх - a 2 U xx - Y 1 9 x - Y 2 P x + №V t + Mx 1 , t ) = 0 ,

• < TZ t ( x, p, t ) + Z p ( x,p,t )=0 ,                                                     ⁽⁸⁾

c9 t + dP t - k6 xx - Y i ^ xt = 0 ,

,d9t + rPt - hPxx - Y2^xt = 0, with the following initial and boundary conditions

u ( x, 0) = u o ( x ) , u t (x, 0) = u 1 ( x ) , ^(x, 0) = ^ o ( x ) , V t ( x, 0) = ^ i ( x ) , 9 ( x, 0) = 9 o ( x ) , P ( x, 0) = P o ( x ) , < z ( x,p, 0) = f o ( x,Tp ) , z ( x, 0 ,t ) = V t ( x,t ) , u (0 ,t ) = v (0 ,t ) = 9 (0 , t ) = P (0 ,t ) = 0 _u x (1 ,t ) = V x (1 ,t ) = 9 (1 ,t ) = P (1 ,t ) = 0

x E (0 , 1) , x E (0 , 1) , x E (0 , 1) , ( x,p ) E (0 , 1) x (0 , 1) ,              (9) ( x,t ) E (0 , 1) x (0 , + ro ) , ( V t >  0) , ( V t >  0) .

Introducing the vector function U = ( u, u _t , z, v, v t , 9, P) ^T . Then system (8)-(9) can be written as

( U ‘ ( t ) = AU ( t ) , t >  0 ,

\

(U (0) = Uo = (uo,ui,^o , vi,fo,9o,Po)T, where the operator A is defined by

/                            Ut u ut ^ ϕt z 9 P

P 1 1 [ a i U xx + a 2 ^ xx ]

ϕ t

A

P 2 [ а з V xx + a 2 U xx + Y 1 9 x + Y 2 P x - mV t - №z( x, 1 , t )] ^- T z p                             ■

( ^ ) 9 xx - ( hd ) P xx + ( r^{Y 1 - dY 2} ) V tx

V         ( ch ) P xx - ( kd ) 9 xx + ( c²²^ ) V tx        )

Now, the energy space is defined by

H = Hi(0,1) x L2(0,1) x Hi(0,1) x L2(0,1) x L2 ((0,1),L2(0,1)) x L2(0,1) x L2(0,1), where

h 1 (0, 1) = f e H ¹ (0 , 1); f (0)=0) .

Let

U = (u,u t ,4,4 t ,z,9,P) T , U = (u,U t ,4,4 t ,z,9,P) .

Then, for a positive constant ξ satisfying

^TЫ < € < ^T <² № - W,

we define the inner product in H as follows

( U U ) H = j [ p r u t u t

+ a i U x U x + p 2 ^ t 4 + а з Ч х Ч х + a 2 x 4 x + 4 x U x) ] dx

1                             11

+

/

[c99 + d^P9 + 9P) + rPP] dx + €

II z ( x,^p)      .. ....

The domain of A is

d (A ) = {U e H | u, Ч e h 2 (0, 1) , u t , ч e h 1 (0, 1) ,

9, P e h1(0, 1), z, Zp e L2(

H2, 1) = {f e H², 1); ffx<1) = o}.

Clearly, D) is dense in H.

We have the following existence and uniqueness result.

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

U e C (R⁺; D<A)) П C¹(R⁺; H) .

<1 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,u_t,4,4t,z,9,P)^Te D), by using the inner product and integrating by parts, we obtain

(AU, U^h

—

k j 9X dx — h j

P_x²dx

—

(^w—2T)/ 0

ϕ_t²dx

—

—

— I z²<x, 1, t) dx.

2T J

Using Young’s inequality, we obtain

—^2 j z 1,t) 4t dx ^ ^22^ j z²<x, 1,t) dx + -^t²- j Ч2 dx-

Therefore, from the assumption (11), we have

{AU, U)h

<

k

θ_x²dx

(^{№ —}2t  V) / ^2 dx

— h j Px2 dx — ^2— ^т^^ / z2(x, 1, t) 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,v,Vt,z,9,P)^T € D(A) satisfying

(Id — A )U = F,

which is equivalent to u — ut = fi,

P1Ut — aiUxx — a2Vxx = Pf2,

V^—Vt = ^f3,

< p2Vt — a3Vxx — a2Uxx — Yi9x — Y2Px + ViVt + №z(x, 1, t) = P2f4,          ⁽¹³⁾

TZ + Zp = Tf5,

X9 — rk9xx + hdPxx — (rYi — dY2) Vtx = Xf6,

^XP — chPxx + kd9xx — (cY2 — dYi) Vtx = Xf7.

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

J ut = u — fi, Vt = V — f3 .

It is clear that ut € H1(0,1) and Vt € H1(0, L).

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

z(x, p, t) = v(x)e ^Tp

—

^ρ f3(x)e^-Tp + те^-тр j e^Tsf5(x,s) ds, 0

clearly, z, z_p € L²((0,1) x (0,1)).

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

piu — aiuxx	— a2Vxx = gi,
nv — a3Vxx	— a2uxx — Yi9x — Y2Px = g2,
X9 — rk9xx + hdPxx — (rYi — dY2)Vx = дз,
XP — chPxx	+ kd9xx — (cY2 — dYi)Vx = g4

, where

П = Р2 + № + №e T, gi = pifi + pi f2,

< g2 = P2f4 + пfз - P2Te-T / eTsf5(x, s) ds, дз = —f6 - (rYi — dY2 )f3x,

,g4 = ^—f7^{- (c}Y2 ^{- d}Yi) ^f3x•

We multiply (16)i by u, (16)2 by (, (16)з by c9, (16)4 by rP, (16)з by dP and (16)4 by d9 and integrate their sum over (0,1) to find the following variational formulation

B ((u, (, 9, P)^T, (u, (, 9, P)T) = G(u, (, 9, P)T,                    (17)

where B : [H^O, 1) x H^O, 1) x L²(0,1) x L²(0,1)]²—> R is the bilinear form given by

B ((u,(,9,P)^T, (u, p,6,P)T) = p_i У uu dx + a_i j

11  1

+ п У (p dx + аз У (_xp_xdx + c j 99 dx + k j 9_x9_xdx +

uxux^dx+a²/■xux⁺ux^{px) dx}

+d

0   00

r j PP dx + hj PxPxdx

У (вР + P9) dx + Yi j (9px - (x9) dx + 72 j (Ppx - (xP) dx,

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

11     1     1     11

T       cr   dd

G uH^pJdP³) = g_iH dx + g₂(5 dx + — дз9 dx + — g₄P dx + — дзP dx + — g₄9dx.

00     0     0     00

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, (p, 9), P) G H(0,1) x H(0,1) x L²(0,1) x L²(0,1), problem (17) admits a unique solution (u, (, 9, P) G Hⁱ(0,1) x Hⁱ(0,1) x L²(0,1) x L2(0,1). The application of the classical regularity theory, it follows from (16) that (u, (, 9, P) G H2(0,1) x H2(0,1) x H0)(0,1) x H0)(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

E(t) =

2/

+ aiux + p2(t + аз(х + 2a2ux(x + c9 + 2d9P + rP ] dx

+ 2

У У z²(x,p, t) dpdx.

Noting (4), we have for 9, P = 0,

c92 + 2d9P + rP2 = -92 + r

(-=9 + VTp)2 > 0,

then we get that the energy E(t) is positive.

The stability result reads as follows.

Theorem 2. Let (u, z,^,9,P) be the solution of (8)-(9) and (4) holds. Then there exist two positive constants k0 and k1, such that

E(t) ^ koe^-k1t(V t > 0).

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

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

11   1   1

dt E(t) ^ -k J 9X dx - h J Px dx - Ci j 00  0

^t dx — C2j z2(x, 1,t) dx ^ 0, where

Cl = ^ - M - A > 0, C2 = — - —>

1    '        2      2-      ’     2    2-

<1 Multiplying (8)1, (8)3, (8)4 and (8)5 by ut, ^t, 9 and P, respectively, and integrating over (0, 1) with respect to x, using integration by parts and the boundary conditions, we obtain dt 2 j (piut + aiuX + P2^t + аз^Х + 2a2UxPx + c92 + 2d9P + rP2) dx 0

11   1   1

= -kj^ dx - hjPx dx - ^1j ^_tdx - ^2 у ^^.(x’ 1’t) dx.

00   0   0

On the other hand, multiplying (8)2 by Tz(x,p,t) and integrating over (0,1) x (0,1), and

recalling z(x, 0, t)

= ^_t, we obtain

ξd                ξ

2 dt / z ⁽x’ P’t) dPdx = 2? ^t ^dx

- 7— j z²(x, 1,t) dx.

A combination of (21) and (22) gives

dt E(t) = -k У 9Х dx - h У P_x²dx -

(^1 - 2?) / ^2 dx 0

• -    2- J z²(x, 1, t) dx 0

• -    p₂ j p_tz(x, 1,t) dx.

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

—^2 j z(x, 1,t)^t dx ^ ^t²- j z²(x, 1,t) dx + y²j ^2 dx.

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

Li(t) = —pi j uut dx, satisfies, for any Ei > 0, the estimate

1                            21

+ E_ij Фх dx.

Li (t) ^ —Pi У u²dx + ^^ai + 4"2”) j

0                          10

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

L1(t)

-ρ1

j u2 dx + a_i

j uX dx + a₂

uxϕx dx,

then, by Young’s inequality, we obtain the result. >

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

L2(t) = aiP2

ϕϕt dx

-

a2ρ1

ϕu_tdx,

satisfies, for any E2 > 0, the estimate

L2(t^ ^ — 2 ^ ^x dx + Ci(e2) ^ ^t dx + £2 ^ ut dx +-----

0   00

₊ ^Ы f px dx + ^x f z²(x, i,t) dx, _{a               a}

θ_x²dx

where

a = aia3 — a2 > 0, Сз(е2) = aiP2 + ^a^^i + a^. a      4е₂

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

L₂(t) = -a j V2 dx + a_ip₂У ^2 dx —

a2Pi j Vtut dx + Yiai jtp9_xdx

+ Y2ai

ϕP dx - µ1 a1 ϕϕ_tdx -

^₂a_i У ^z(x, 1,t) dx, 0

where a = „301 established. >

-

a? > 0. Using Young’s and Poincare inequalities, estimate (26) is

Lemma 4. Let (u, z, y, 6, P) be the solution of (8)-(9) and (3) holds. Then the functional

L3(t) — a^lP2 [ ytudx — a2

a3ρ1

u_tϕ dx, a2

satisfies, for any £3 > 0, the estimate

L3^(t)<^—ai у u? dx⁺

„3 /yX dx + 2017? t₆d dx . - • P. dx a₂                a₂

+ C2(£3) У y? dx +

a¹^² f z²(x, 1,t) dx + _a2₂

ε3   u_t²dx,

where

C4 (£3) = 2

a²₂

1+ -—

4ез

a1 ρ2

-

a2

a3ρ1    _.

<1 By differentiating L3 (t) with respect to t, using the equations (8)1 and (8)2, and integrating by parts, we obtain

L3(t) — —ai J* Ux dx + „3 J*

yX dx + Y1a1

-

µ1a1

ϕtu dx a2

a2

/6Xu dx + Y2 1   Pxu dx a2

-

P?a1 У z(x, 1, t)u dx + ^ a1P2

-

a3ρ1

a2

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

Lemma 5. Let (u, z, y, 6, P) be the solution of (8)-(9) and (3) holds. Then the functional

L4(t) — У У е-2трz2(x,p, t) dpdx, satisfies, for some positive constants n1 and n2 , the estimate

L4(t) C —n1 j j z(x,p, t) dpdx —

n2 j z(x, 1,t) dx +—У y? dx.

<1 By differentiating L4(t) with respect to t, and using the equation (7)з, we obtain

L4(t) - 2 У J e

2^Tpz(x, p, t)zt(x, p, t) dpdx — — У У e^-2^Tpz(x, p, t)z_p(x, p, t) dpdx

- — M dh (e

2^Tpz²(x, p,t)) dpdx — 2 У У e^-2Tpz²(x, p, t) dpdx

<

-n1

z²(x, p, t) dpdx

-

n2

У z²(x, 1,t)

dx +— У p2 dx, 0

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

L(t) :— NE(t) + ^ NiLi(t) + L4(t), i=1

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

L‘(t) ^ — [piNi

- ε2N2

-

E3N3] у

u_t²dx

-

т N^3—(^a1⁺41) ^N1

u²_xdx

CiN ^—C3⁽E2^)N2^—C4⁽E3^)N3--

τ

ϕ_t²dx -

[2 N2 ^{— a}3^N3^—E1N1] У Vx ^dx

kN — 22^ N2 — ^2a1Yl N3] Z @x dx — [hN — 2^ N2 — 2^^ N3I Z Px2 dx a            a²₂            ^x                  a            a²₂            ^x

-

ni J j z²(x,p,t) dpdx — [^N + n₂

-

a

2a1p2 a²₂

At this point, taking Ei — 1/Ni, i — 1, 2, 3. We then choose Ni large enough so that Ni > 2/pi. After that, we select N3 so that a1              a22N1

— N3 ^—( ai +4— JNi > 0.

Then, we choose N2 large enough so that j N2 — a3N3 — 1 > 0.

Finally, we select N large enough so that

C1N - C3(E2)N2 - C4(E3)N3> О, τ kN - 2аЫ N - 2 N > о, aa hN - -'   ■ N - 20172 N, > о, aa

C2N + П2 - 2    N2 - 2 '2 N, > 0.

aa

Consequently, from the above, we deduce that there exist a positive constant α0 such that

L‘(t) < -aoE (t).

On the hand, it is not hard to see that L(t) ~ E(t), i. e., there exist two positive constants ai and α2 such that

• a1E(t) ^ L(t) ^ a2E(t) (Vt ^ 0).

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

L‘(t) ^ -kiL(t) (Vt > 0), where ki = aa2 • A simple integration of (31) over (0,t) yields

L(t) ^ L(0)e^-k1t (Vt ^ 0).

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