Decay rates of solutions to thermoviscoelastic plates with memory

MA Journal of Applied Mathematics (1998) 60,263-283

Decay rates of solutions to thermoviscoelastic plates with memory

JAIME E. Mur;joz RIVERAt National Laboratory for Scientific Computation, Department of Research and Development,

Rua Lauro Miiller 455, Botafogo Cep. 22290, Rio de Janeiro, RJ, Brasil AND

RIOCO KAMEI BARRETO University Federal Fluminense, Rio de Janeiro, RJ, Brasil

[Received 13 March 1997 and in revised form 7 August 19971

We consider the thermoelastic plate under the presence of a long range memory. We find uniform rates of decay (in time) of the energy, provided that suitable assumptions on the relaxation functions are given. Namely, if the relaxation decays exponentially then the first order energy also decays exponentially. When the relaxation g satisfies

--c&)r+“P < g’(t) < -~g(t)‘+“~; and g, g’-“P E L’(R) with p > 2

then the energy decays as l/( 1 + t)P. A new Liapunov functional is built for this problem.

1. Introduction

In this paper we study the thermoelastic plate model endowed with long range memory. Our attention will be focused on the asymptotic behaviour of the solution as time goes to infinity. More precisely, let us denote by 52 an open bounded set of Iw* with smooth boundary lY We assume that the boundary is divided into two parts, such that

r = ro U rl with TO nF1 = 0 and ro # 0. (l-1)

Let us denote by v = ( ~1, ~2) the external unit normal to r, and by q = (-~2, vi) the unitary tangent positively oriented on r. The equations which describe small vibration of a thin homogeneous, isotropic thermoviscoelastic plate of uniform thickness h are given by:

s

t

utt - hdu,, + A*u + ad0 - g(t - r)A*u(t) dt = 0 in 52x10, oo[, (W 0

et - KAO + y0 - aAut = 0 in Qx]O, OO[, (l-3)

u(x9 y, 0) = Uo(& y>; u&L y9 0) = uI(x, y> (x9 Y> E 52 (1.4)

t Also at: Institute of Mathematics, Federal University of Rio de Janeiro, RJ, Brasil.

@ Oxford University Press 19%

264 JAIME E. MUfiOZ RIVERA AND RIOCO KAMEI BARRETO

with the following boundary conditions:

dU u=- = 0

dV on rox]O, oc[, (13

(s

t

l31u+ae 431 g@ - r)u(t) dt = 0 on ri x10, oo[, (W 0

autt ae

(s

t &~-h~+~~-a, o la- t)u(t> dt

I = 0 on ri x10, oo[, (1.7)

ae G + he = 0 on rx]O, oo[, WV

dAu &U = Au + (1 - ,u)Blu, &u = -

aB2u av +(l -P>-

ar7 ’

and B1 and B2 are given by

a2u 0 = 2vp2-- -

,a224 2 a2u axay ‘1 ay2 - ‘2 ax2 ’

B2u = (v; - v,“) &+v1v2{$-3

In (1.2), u = u (x, y, t) denotes the position of the plate. We may interpret equation (1.2) as saying that the stresses at any instant depend on the complete history of strains which the material has undergone. By g E C2(Iw, lEX> we denote a positive real function and the constant ,X is assumed to be in the interval IO, i[. The constants K, y, a, h, and h are assumed to be positive.

The main result of this paper (Sections 3 and 4) is the uniform rate of decay of the solution to the system (1.2) to (1 .S). We prove that the rate of decay of the solution depends on the rate of decay of the relaxation kernel g. That is, when g decays to zero exponentially then the vertical deflection and the thermal difference also decay to zero exponentially. Moreover if the relaxation decays polynomially then the vertical deflection and the thermal difference also decay polynomically. This shows that the memory effect is strong enough to produce the dissipation which enables us to show that the energy decays uniformly (exponentially or algebraically).

Let us mention some other papers related to the problems we address. Dafermos (1970) proved that the solution to viscoelastic systems goes to zero as time goes to infinity, but without giving explicit rate of decay. Lagnese (1989) considers the linear viscoelastic equation, obtaining uniform rates of decay but introducing additional damping terms acting on the boundary. Uniform rates of decay for the solutions of linear viscoelastic equations with memory were obtained recently by Mufioz Rivera (1994).

Concerning the asymptotic stability of the solution for inhomogeneous anistropic thermoelasticity we have the pioneering work of Dafermos (1968). In that work it is proved that the solution is asymptotically stable as time goes to infinity. In particular that the thermal difference as well as the rotational part of the displacement goes to zero; but no rate of decay is shown. For 3-dimensional isotropic homogenous materials the work of Dassios &

DECAY RATES OF SOLUTIONS 265

Grillakis (1984) shows that a displacement vector field can be decomposed into two parts, the rotational and selenoidal components. The rotational component decays uniformly to zero like t-5 while the selenoidal component conserves its L*-energy. This result was improved in (Mufios Rivera, 1993). The above results illustrate that the displacement, in general, does not go to zero. This because for n-dimensional thermoelastic bodies, the displacement is a vector-valued field and the dissipation given by the thermal difference has only one degree of freedom. By contrast, in the l-dimensional case, both the thermal difference and the displacement have the same degree of freedom. So, uniform rate of decay is expected as was shown in (Hansen, 1992; Henry et al., 1993; Kim, 1992; Mufioz Rivera, 1992; Renardy, 1993). For plates the situation is similar to the case of l-dimensional thermoelasticity, because the vertical deflection as well as the temperature are scalar functions. So, uniform rate of decay is also expected and was proved in (Avalos & Lasiecka, to appear; Kim, 1992; Liu dz Renardy, 1995; Mufioz Rivera & Racke, 1995; Mufioz Rivera & Shibata, 1997).

Adding the viscoelastic damping to the thermoelastic system, we should expect that the elastic and thermal parts of the energy continue to decay to zero exponentially even if the relaxation kernel decays polynomially, but this is not true as we can see in Re- mark 1.1 below. So, the memory effect prevails over the thermal difference and introduces a new difficulty to the problem. Unfortunately the method used to obtain uniform rates of decay in the above works are based on second-order estimates, which make the problem of securing estimates more complicated. Thus, the methods that have been used for estab- lishing uniform rates of decay fail in the case of a thermo-viscoelastic system, and so new asymptotic techniques have to be devised.

The method we use here is based on the construction of a functional C for which an inequality of the form

d zL(t) < --cc(t)‘+;

holds, with c, p > 0. Let us describe briefly all the sections of this paper. In Section 2, we prove the existence

of weak solutions. Furthermore, we show some regularity results. To do this we assume that g satisfies

s

00 g, g’, g” E L’(0, OQ), 1 - g(t) dt > 0.

0

g(t) 2 0, g’(t) < 0. (1.10)

Since the methods of Section 2 are quite standard we will only give a brief summary of the procedure. In Section 3 we show the exponential decay of the energy assuming that g satisfies additionally

-cog(t) < g’(t) < -c1g(t), 0 < g”(t) < c*g(t)* (1.11)

Finally, in Section 4, we show that the polynomial rate of decay of the relaxation implies

266 JAIME E. MUNOZ RIVERA AND RIOCO KAMEI BARRETO

the polynomial rate of decay of consider the weaker hypotheses

the solution. To do this, instead of assumption (1.11) we

-cog l+l'p(t) < g'(t) < -C*g'+'lP(t), 0 < g”(t) < c2g1+1’p(t), p > 2, (1.12) 00 g

‘-*‘P(t) dt < 00, p > 2. (1.13)

Assumptions (1.12) and (1.13) mean that g m (1 + t)-P for p > 2. All the above constants Ci, i = 0, 1,2, are positive.

REMARK 1.1 It is well known that the solution of the thermoelastic plate system decays exponentially as time goes to infinity. But when we introduce the viscoelastic dissipation with relaxation kernel g satisfying assumptions (I. 12) and (1.13), the solution does not decay exponentially any more. That is, the memory effect given by the convolution term prevails over the thermal dissipation. In fact, to facilitate calculations let us assume that the system (1.2) to (1.4) satisfies the following boundary conditions:

U = Au = 8 = 0 inr.

It was proved in (Mufioz Rivera & Racke, 1995) that the solution of the thermoelastic plate system (without memory) decays exponentially as time goes to infinity. Let us supposethat the same is true for the thermoviscoelastic system with the kernel satisfying assumptions (1.12), (1.13). Take uo = WI, ut = wi and& = ~1, where wi is the first eigenfunction of the A operator with Dirichlet boundary condition. Denoting by hl the first eigenvalue we conclude that the solution can be written as

u(x, 0 = f(Wl and m, 0 = w)~l,

where f and h satisfy

s

t

(1 + h&f” + h;f - ahlh = A; g(t - t) f (t) &, 0

h’+Khlh+Alyh+ahlf’=O.

Let us rewrite equation (1.2) as

s

t

utt - hAutt + A2u + aAt3 = g(t - z)A2u(t) dt in 52x10, oo[. (1.14) 0

Since the elastic and thermal parts of the energy decay exponentially, it is easy to see that the left-hand side of the above equation also decays exponentially. While the right-hand side satisfies

s

t

g(t - t)A2u(t> dt = 0 s

t

g(t - z)f(z) dt A2w1. 0

l -

. - I


Note that f(z> = o(e-yr), so we have t I 5% s g(t - t)emyr dx =

0 s 2 g(t)e-yct-‘) dx = emyf ‘g(t)eYr dx.

0 s 0

It is not difficult to see that when t + oo

s

t e-Yt g(t)eyr dx = o((1 + t>-P>.

0

But this is a contradiction because the left-hand side of (1.14) decays exponentially. This proves that the exponential decay is not expected.

2. Existence and regularity

In this section we study the existence as well as the regularity system (1.2). To do this we introduce the following spaces:

LetOct+;we define the bilinear form a (9, l ) as follows:

V := { 2) E H’(D); 2) = 0 on ro) ,

( 3W W := wEH*(Q);w=;=Oon~0

I .

of weak solutions of

a*u a*v a*u a*v a224 a*v a(u,v)= --+--+p

ay* ay* --+-- ax* ay* ay* aX*

2 2

+ 2(1 -/.&au axay axay

Thanks to Korn’s inequality the bilinear form a(*, l ) defines a norm in W equivalent to the usual norm of H*(Q). Note that the space V together with the norm

is a Hilbert space. Frequently, we shall use the Green’s formula for A* which is proved in (Duvaut & Lions,

1976). For proof we reproduce the proof here.

LEMMA 2.1 Let u and v be functions in H4(i2) n W. Then, we have

s

2 av (A u)v dA = a@, 21) + (B2u)v - (&u)- drl

L? au I cw

Pro@ From Green’s formula we get

268 JAIME E. MUROZ RIVERA AND RIOCO KAMEI BARRETO

s 2 (A u)v dA= ilAU

(- 52 s f, av

)vdrr-- a,;drl+/- s

Au Av dA l-1 $2

- -

s ( *)v drr -

l-1 av s l-1

Au; dI-i +a@, v)

s

a2u a2v a2u a2v +(1-p) --+$-ay2dA-2(1-p)

s a2u a2v

a a3 a3 -- dA. Q axay axay Recalling the definition of Br and B2 and using

s a2u a2v a2u a2v a2u a22, -- 52 ax2 a9

+ --dA-2 ay2 ap s

--dA= sz axay axay

our result follows cl

To show the existence of regular solutions we first prove, using the Galerkin method, the existence of weak solutions (see Definition 2.1 below), then using elliptic and parabolic regularity we will obtain the regularity of solutions to the thermoviscoelastic plate equation. To simplify our analysis, we introduce the following binary operators:

s t g 0 a224 = g(t - ee40 - U(t), u(t) - u(z)) dL

go”u=~tg(t-r)~ 2 lVu(x, t) - Vu(x, t)l dAdt,

With this notation we have the following.

LEMMA 2.2 For any v E C*(O, T; H2(L?)) we get

s t a( g(t - t)v(t) dt, v,) = 0

-ig(t)a(v, V) + igf 0 a22,

-Tdt goav ‘“( 2 -(~+w], t s(t - t)Vv(t) dt l Vv, dA =

ss

t go - t)v(x, t) dtv,(x, t) dA =

In 0

DECAYRATESOFSOLUTIONS 269

Pro@ From the symmetry of a (0, 0) we get

d dt {g cl a2v} = g’ cl a2v - 2

s

t t go - @Wt>, v(t)) dt + 2

s g(t - t) dt a(v, v,)

0 0

s

f

=g’oa2v-2 d t

go g(t) dt a(v, v> 0

- t>aW), vt(t>> dt + dt (s 0 I

-g(OO, v>.

Which shows the first identity. The proofs of the others identities are similar. cl

The definition of weak solution we use in this work is given as follows.

DEFINITION 2.1 We say that the couple (u, v) is a weak solution of equations (1.2) to (1.8) if

u E C’([O, T]; V) n CO([O, T]; W), 8 E CO([O, T]; H’(52))

and the following identities are satisfied:

T

ss

T T

-utqt - hVu l VP, dA + 0 n s

a@, p)dt +a 0 ss s

T

BAcpdxdt- a(g*u, p)dt 0 i-2 0

- - s u&, 0) dA + h s VulVspL 0) dA L? A-2 T

ss

T - 6+kt dA dt +

0 52 ss

T

KV~V+ + ye@ dAdt + a ss

Au@, dAdt 0 L? 0 In

- -

s

T

Oo+(*, 0) dA - a s

Auolc/(*, 0) dA - h ss

9@ drdt, 52 52 0 l-1

WI

for any function cp E C’([O, T]; W) such that (p(*, T) = 0, p,(*, T) = 0 and @ E C’([O, T]; V) such that @(-, T) = 0.

Note that since & # usual Sobolev norm I] l 1

0, Korn’s lemma implies that ,/m is a norm equivalent to the (2 on W. Let us introduce the energy function

2 + h]Vvt12dA + (1 - s 0

‘(gdt)a(v, v) +g 0 a2v +ll+12dAJ ;

let us denote by (wi E W; i E Af) an orthonormal basis of W. In these conditions we able to show the existence of weak solution to the thermo-viscoelastic plate equation.

THEOREM 2.1 Let us suppose that g satisfies (1.9) and (1.10). Then for any initial data (UO, u 1 , 60) E W x V x L2(s2), h > 0 and T > 0, there exists a weak solution to equation (2.2).


Pro@ Our starting point is to construct the Galerkin approximation urn of the solution,

m

urn(-, t) = c hi,m(t)wi(‘), e”(*, t) = ~k,.m(t)s(*),

i=l i=l

which is given by the solution of the approximated equation

s uzwjdA+h s

VuzVWj dA + a(um, Wj) - a(g * Urn, Wj) + a s

8”Awj dA = 0, 52 52 i2

(23

s 8,FVj dA + K s

VBmVvj dA + Y BmAvj dx +a Auyvj dA = -h emvj dr. 62 In s sz s 52 s r

(24

um(*9 O) = UO,m, Uy(‘, 0) = Ul,m, 0”(*, 0) = 00,m

UO,m = Ul,m = ulwi dA I Wi 9

Let us denote by A = (aij) the matrix given by

Uij =

(s

wiwj dA+h s

Vwi l Vwj dA . Lz sz >

We can easily verify that A is a positive definite matrix, so the existence of the approximated solution urn is guaranteed. Let us multiply equation (2.3) by h) m, equation (2.4) by kj m then summing up the product result in j and using Lemma 2.2 we conclude that: 9

]u~12+hlVu~~2dA+ 1 ( -~gd~)a(um,um)+g~~2um+~~~~2dA]

= ig’ cl a2um - ig(t)a(u”, urn) - K s

IVBm12 dA - y In s

10” I2 dA - h 10” I2 dr. L? s r

In view of the hypotheses on g we get

E(t, Urn, em) < E(0, Urn, em);

and from our choice of ~0,~~ and u1 ,nz it follows that


Urn is bounded in C([O, T]; W) n C’([O, T]; V),

em is bounded in C([O, T]; L2(Q)) n L2([0, T]; H’(Q)).

Multiplying equation (2.3) by /3 E C2([0, T]), such that p(T) = 0 and integrating over [0, T] we have

T -

ss Uywj/& + hVuy l VWjbt - aedwjp dA dt

0 i-2 T

+ s

a(um, wj)B dt - 0 s

T

a(g * urn, wj)p dt 0

- - s

ul,mwjp(O) dA + h 52 s

In Vul,mVwj~(O) dA.

Letting m + oo and using the density of the terms { WjB; j E N, /I E C([O, T])}, we get that u satisfies (2.2). Similarly we get that 8 also satisfies the weak formulation. The proof is now complete. cl

To prove the regularity of the solution we introduce the following definition.

DEFINITION 2.2 We say that (UO, ~1, eo) is k-regular if

u j E H2+k-j(S2) n W, j = 0,. . . , k, uk+l E V, Bj E Hk-j(52),

where uj and 8j are obtained by the following recursive formula:

Uj+2 - hAuj+2 = A2uj - ad6j - f,,(O),

8. J+l - KAej + y8j = aAuj+l,

au * J+2 h- ae

3V = B2Uj + a-onrt,

tlV

Mj x + Aej = 0 onr,

and also

aUj

ui = av = 0 on ro, 44 =0 on rr, Vj = l,-,k,

where fj is given by

j-0 := 0,

. f j+l = fi(t> - g(t)A2uj, fj E LYO, T; L2(Q)), j = 1, l l * , k.

To show the regularity result we will use the following lemma.


LEMMA 2.3 Suppose that f E L2(i2), g E Hi (Ii) and h E Hs (l-i); then any solution of

d-b w> =~fwdA+~l~wd~i+~lh~drr VWEW

satisfies

v E H4(J2)

and also

A2v= f ina,

3V VC--

au = 0 on PO,

&v = h, &v = g on&

Moreover, if (f, g, h) E Hk(i2) x Hk+1/2(I’l) x Hk+3/2(l-‘l), then v E Hk+4(i2).

Pro05 See (Lions & Magenes, 1972).

The regularity of the solution is established in the next theorem.

cl

THEOREM 2.2 Let us suppose that the initial data (~0, u I,&) is k-regular (k > 2) and hypothesis (1.1) holds. Then there exists only one solution of equation (1.2) satisfying

U E c’([o, T]; v n Hk+‘(i2)) n C’([O, T]; w n Hk+2(Q)),

0 E c’([O, T]; Hk(52)),

and also

u E Cj([O, T]; V n Hk+2-j(i2)), 8 E Cj([O,T]; Vn Hk-j(Q)), J c0,o.o~.

Proo$ Let us take

VO = uk+l, W I= uk+2, 00 := ek+l, f := fk+l*

From Theorem 2.1 there exists only one weak solution (v, #) of equation (2.2) satisfying

v E c’([O, T]; V) n C’([O, T]; w), 0 E c’([O, Tl; L2(a)) n L2([0, T]; v).

Let us write

s 11

v(t) dt dtl 0 l l

0

It is easy to see that


* E ck+‘uo, n V) f-l Ck([O, T]; W); 8 E P([O, T]; L2(i2))

and satisfies equation (2.2) for u(O) = ~0, ~~(0) = ur and f = 0. Taking I.J = ~0 with 8 E Cr(O, T), w E W in equation 2.2 and q = it with z E C,“(O, T), w E V, integrating by parts with respect to the time variable and using the du Bois-Reymond lemma, we conclude that

a(u, w> - a(g * u, w) = - s (utt - yAutt + aA0)w dx - si au,, ae

sz l-1 y av - - czz w dlj.

I

To prove the regularity result we reason by induction. Let us suppose that k = 2. From the parabolic regularity we conclude that

8 E C(0, T; H2(1n)).

So, Lemma 2.3 implies that u - g * u E H4(Q), and also satisfies the following boundary conditions:

&u--y~+ch5-+2g*u=0 on&

Using the resolvent equation, it follows that u E H4(i2). So, the result is valid for k = 2, which means that u satisfies (1.2) to (1.7) in the strong sense. Differentiating (k - 2)-times equation (1.2) to (1.7) we get:

(k-2) ’ $’ -hA (f +A2 (ki2) +aA 0 -

s go - t)A2

(k-2) U (t) dt= fk in 52 x10, oo[,

0

(k-1) (k-2) (k-2) 8 -/cd 8 +y 0 -aA u (k-I)=0 inQx]O, oo[,

(k-2) a (ki2) u =- =0

ih on ro x10, m,

(k-2) W-2) t a 8 +6 u 4%

(s go

W-2) -t) u (0, t)dz = 0

I on G xl09 m,

0

(k-2)

fJ2 (k;2) -y a? ae t =+a-- x32

(s so

(k-2) av -t) u (9, t)dt on fi xl& cd,

0

(k-2) a 8 k-2 av+A e=O on rx]O, oo[.


Using the resolvent Volterra equation we have that (k-2)

u satisfies

(k-2) A* u = F,

(k-2) d (kii2) u =-

av =0 onro,

(k-l) * x31 (ku2)= a 8 a

s

(k-1) r(t - t) 8 dt onri,

0

(k-11 a? a0

(k-1) x3* (ku2)= y- - a

av -+ s

t a %’ a 0 r(t - t)x - a - dt

aP 0 a,cc vG

l -

. -

where Y is the Volterra kernel associated to g and

- (ii’ (*, t) + A ‘II’ (0, t) + aA (k-1) 0 +fk.

From hypothesis (1.1) and the elliptic regularity given in Lemma 2.3 we

on r1,

dt

onclude that

% C(0, T; H*(C?)) (k-2) (k-2) + u E C(0, T; H4(52)) and 8 E C(0, T; H*(Q)),

'ki2'~ C(0, T; H4(i2)) (k-4) + ‘kU4& C(0, T; H6(s2)) and 8 E C(0, T; H6(D)),

'ki4'E C(0, T; H6(i2)) ‘kU6’~ C(0, T; H8(Q)) and W-6) + 8 E C(0, T; H’(C)).

Reasoning by induction we get

(k-2 j) u E C(0, T; H*'+*(n)) e u E Ck-*'(O 9 T; H*'+*(Q)).

Using the intermediate derivative theorem (see (Lions & Magenes, 1972, Theorem 2.3, p. 15 and Theorem 9.6, p. 43)) our conclusion follows. Uniqueness is immediate for k- regular solutions (k > 2). The proof is now complete. cl

3. Asymptotic behaviour: Exponential decay

In this section we study the asymptotic behaviour of the solution of (1.2) to (1.7). Note that as deduced in Section 2, the energy function satisfies

d zE(t, U, e) = ig’ I a*u - ig(t)a(u, U) -K

s IVBl’dA-y

s lel*dA -h 18l*dr.

52 sz s r (3.1)


To prove exponential decay, first write w = u - g * u. It follows that w satisfies

wit - hdw,, + A2w + g’(O) {u - hdu} + g(0) (ut - hdu,) + g” * {u - hdu} (W

+aA0 = 0 in Dx]O, oo[,

6 - Kde - dUt = 0 in Inx]O, oo[, (33

w9 y, 0) = Uo(& y>, wtk y9 0) = Ul(% Y) - gWuo(L yL (3.4

with the following boundary conditions:

dW WC---=(-)

au on GGw, a, (33

B*w +ae = 0 on ri x10, oo[, W)

awt au, au af * u ae W-J - hx + g(O)h~ + g’(o)h,v + h7 + “ay = 0 on rl x]O, oo[, (3.7)

ae z + he = 0 on rx]O, oo[. 68)

Next we introduce the new energy function associated to system (3.2) to (3.8),

6

Iw,12 + hlVw,12 + lAv12 dA + x W),

i=l

where Si (t) is given by

&(t) = - ~h(~[~Vu]2+(~gdr)]Vu]2-2(g*Vu)Vu] dA-hgDVu],

Sdt)=T (so [g~O)iul2+ ([or) lul2] dA 3’ 0 U})

s,o=~h(~g~O,[~V,lz+(~g’d~) ivui2] dA-g’nvuJ.

s,(t)=-; (8” clu+l [/g’*ul2- (pr) lulj d+

s,(t)=-;h (p” q vU+s, [Ig’*vu12- (I’fdr) /vui2] ,,)*

The point of this study is to establish an energy inequality given in the next lemma.

LEMMA 3.1 Under the above conditions, the solution of system (3.2) to (3.8) satisfies the following inequality:

276 JAIME E. MUtiOZ RIVERA AND RIOCO KAMEI BARRETO

d ,w < cc 1s

g(t)IVu12 dA + g •I Vu + s2

s, WI” ,A)

+C (g(t>a(u, U> + g I a2u} + E s

IVu12 dA - g(0) s

lut12 + hlVu,12 dA. 52 52

Pro~$ Let us multiply equation (3.2) by wt, applying Green’s formula and using the boundary conditions we get

Izo,12dA+h s 52

P

Vwt12 dA + a(w, w)

f - g’(O) I uwt + hVuVw, dA -g(O)

I utwt + hVu,Vw, dA

62 s2 L / \ 2 Y Y

:= I1 + 12 := 13 + 14

- s (,,, * u + hg” * Vu) wt dA 52

-h awtt aw

- - d(o)av u-4 - g(0) av 2) wtdr -11 [(&w)wI - (&w)%] dr.

yO

l -

. -

Now we will consider each term Zi. Using Lemma 2.2 we get

g’(O) d h(t) = -2dt

s 52 M2 CM - g’(o); / (g * u)u dA + g’(0)

s (g * u)ut dA

lu12 + g * uu + Q1 (I’pdr) lu12] d,“- ;g au] 5

:= sl(t) I

+g (0) duagw Tg’ou- 2

s lu12 dA.

r(2

Similarly we get that

d 12(t) =

g’(o) $(t) + hTgt •I Vu -

g’(o)g(t) h 2

s IVu12 dA.

L2

Now we will consider 13:


13(t) = -g(O) s

g(O)uu, + (g’ * u)u, dA sz

-- - g(0) S L?

:= i3(t)

g(O) +- 2 g” I u -

g mm 2 S lu12 dA.

In

By symmetry we get

d 14(t) = --&94(t) - gum

s lOuI dA.

62 lVul12dA + Thg” •I Vu - g(“)Zg’(t)h /

52

Finally we consider 25:

I5(t) = - S g” * uut - (g’ * u) + g(O)u(g’ * u) - g(O)u(g" * u) dA l2

- s s u ‘{g(0)gN(t - t) - g’(O)g’(t - z)} (u(t) - u(t)} dt dA sz 0

-W)g’(t) - s’mw S Id2 dk 52

analogously

16(t) = d

-&s6(t) - h(g(O)g”(t) - g’@)g’(t)) S lvu12 dA

-hlVu*(l Q {g(O)g’(t - t) - g’(O)g(t - t)} {Vu(t) - Vu(t)} dt dA.

From the above inequalities, Poincare’s inequality and the hypotheses on g we obtain

Ii(t) < d

zSi(t) + CE g(t)(Vu12dA+g •I Vu +c I s

PI dA, i2

for i # 3,4, and

13(t) = d

$53(t) - g(O) ~lutl~+C(~OU+~~f~~lul~dA],

d 14(t) = $940) - g(W

s put I2 + c g El vu + g(t)

52 s, IWdA} l


Finally, we will estimate the term Zo:

Z()(t) = -a s u,AO - g’* uA8 - g(O)Ak dA Q

=a s

vevut + veg’ * vu + g(o)vevu dA sz

< G (s IVei*dA+g q au +S lVutl*dA. sz I s In

So, from the last inequality our conclusion follows. cl

REMARK 3.1 It is not difficult to see that there exists a positive constant C such that

E(t) < CE(t).

To prove it we will show that there exists a positive constant c such that

s (wt12 + hlVw,l* dA + a(w, w) < cE(t). r(2

We only prove the inequality

s Iwt12 dA < cE(t);

sz

the others are similar. Note that

s

t

wt = ut - mu - g’(t - z){u(*, t) - ~(0, t)} dt. 0

From this it follows that

s Iw,(*dA < c (s

I4 I2 dA + g(t) s lul*dA+g •I u .

sz 52 sz J

Using Kern’s inequality our conclusion follows.

Let us introduce the functional

J(t, u) := s

utu + hVu,Vu dA. 52

Now we are in conditions to prove the exponential decay of the solutions.

THEOREM 3.1 Suppose that the initial data (~0, u 1) is 2-regular and the kernel g satisfies conditions (1.11) and (1.9) then there exist positive constants ~0 and ~1 such that

E(t, u, v) < KoE(O, u, v)e-?

DECAYRATESOFSOLUTIONS 279

Proofi Multiplying equation (3.2) by w, integrating over Q and using the boundary conditions we get

d J(t) dt =Q s

lut12 + wut

-a s

A8udA r(2

< s

l&l2 + Wk Q

t 2dA - a(u,u)+a go - t

2dA - (l- c)a(w,w)+C, s 52

r, P 0 +& (I’Sdr)g cl Pu -J, IAvl”dA.

Using Lemma 3.1 we find that

d g(O) dt NE(t) + E(t) + 2 J(t) I

u(t) dt, u >

WV

WI2 dA

(3.10)

(3.11)

I

l - . - at>

< --Ko (s lut12 + hph,12 + lOI2 dA + a(u, U) + g I a2u . sz

:= N(t)

It is not difficult to see that

@N(t) < C(t) < NN(t),

for N large enough. From the last two inequalities our conclusion follows. cl

REMARK 3.2 Note that h > 0 has no role in the above estimates, so a similar result holds for plates with thickness h = 0.

4. Polynomial decay

In this section we will show that when the kernel g decays polynomially then the first-order energy also decays at the same rate. To do so, we consider hypotheses (1.9), (1.12), (I. 13). To prove the main result of this section we will use the following lemma.

LEMMA 4.1 Suppose that g and h are continuous functions satisfying the conditions

CL l+l’q(O 00) n L1 (0,oo) for some 4 > 1 and gr E L1 (0,oo) for some 0 < r < 1; then we have ;hat

s t I&t - wwl dt < is t q/(4+1) IgO - t>l l+cl-r)‘qlh(~)I dt

0 0 I l/(q+l)

X IgO - ~>l’lfwl dt l


Pro05 For any fixed t we have

s

t t Ig(t-t)h(t)ldt =

Sk Jg(t - t) Irl(q+l) Ih@) 1 lbl+l) Jg(t _ t) 1 l-r/(q+l) lh@) Id(q+l) dt.

0 0 /L / Y v l -

u l -

. - . - V

Note that u E LP(O, 00), v E LP’(0, 00), where p = 4 + 1 and p’ = (4 + 1)/q. Using Holder’s inequality, we get

/

t

lg(t - wml dt < 0 v

ot lg@ - wfwl~~ I

WI+*)

(s

t

I

q/(4+*) X I& - Gl l+(l-r)‘qlh(t)ldz .

0

This completes the proof. cl

LEMMA 4.2 Let us suppose that z E C(0, T; E?*(Q)) and g is a continuous function satisfying hypotheses (1.12), (1.13); then for 0 < r < 1 we have

while for r = 0 we get

(s t

g cl a22 < 2 llmll~ dt + 0

V(1+(1-r)p)

1 g 1+1/p I a*2} (1 -r)p/(l +u -4p)

9

ll(p+l)

IIW 11; I

I g 1+1/p c3 a2dp’(1+p) l

Pro05 From the hypotheses on z and Lemma 4.1 we get

s

t

g c3 a22 = g(t - t) a(z(t) - z(z), z(t) - z(t)> dt 0 L v /

:= h(t) t

< (s

s’(t - t)h(z) dt 0

) llw-r)p+l) ( Jd’ gl+l,p(t _ t)h(t) dr}(l-r,rlirl.-nuii,

G { gr I a*z} l/((l-r)P+l) { gl+l/p I a*~}(l-‘)Pl((l-r)P+l) . (4-l)

For 0 < Y < 1 we have

s t gr I a22 = g'(t - M(z(t) - z(t), z(t) - z(Wr

0

s

t

a g’@) 0

dtIkli& 7-+2)* , 7

From here the first inequality of Lemma 4.2 follows. To prove the last part, let us take r = 0 to obtain

I I a22 = I

a(z(t) - z(t), z(t) - z(t))dt 0

s

t

< wml~ + 2 IIW 11; dt* 0


Substitution of the above inequality into (4.1) yields the second inequality. The proof is now complete. cl

From the above lemma we get

g q azu < Co (gl+l/p ( a2U)(l-r)pl(l+(l-r)p) (4-2)

forO<r c 1.

LEMMA 4.3 Let us suppose that f is a non-negative C1 function satisfying

f’(t) < -kdf wll+l’P + (1 +k;)p+l

for some p > 1 and positive constants ko and kl. In these conditions, there exists a positive constant cl such that

/( > P

f (0 G (f (0) + 2h 1 + c’{ f (0)p . P

Pro05 Let us denote by h(t) and F(t) the functions defined by

h(t) := 2Kl

p(1 + t)P’ F(t) := f(t) + h(t).

So we have

F’(t) = f’(t) - 2Kl

(1 + t)p+l

< -Ko[f (t)]l+l’P - K1 (1 + t)p+’

1+1/p [f (t)]l+‘lP + +[h(t)]l+l’p

2KOK’

From here it follows that there exist a positive constant c for which we have

F’(t) < -c {[f (t)]‘+“P + [h(t)]*+‘lp} < -c[F(t)]‘+‘lp,

which gives the required inequality. cl

LEMMA 4.4 Under the above conditions, the solution of system (3.2) to (3.8) satisfies t’: the following inequalil t7

d ZW) G c,

is s(t) I’

52

+c {g(t>a(u

vu12 + JV0i2 dA + gl+‘lp •I Vu

u) + g’+‘lp ~1 a2u} + E s

IVul dA - g(0) s

)u,12 + hlVu,12 dA. Q 52


ProoJ: The only difference with respect to the proof of Lemma 3.1 is to estimate the following term:

- s s u ‘(g(o)g”(t - t) - g’(O)g’(t - z)} {u(t) - u(t)} dt dA. In 0

All other estimates follows using the same argument and the hypotheses on g. Writing

h(t - t) = g(O)g”(t - t) - g’(O)g’(t - t)}

we have that

Ih( < cg’+“P

from which our conclusion follows. cl

We are now in condition to prove the main result of this section.

THEOREM 4.1 Let us suppose that the initial data (UO, ur ) is 2-regular, and that (1.9), (1.12), (1.13) hold; then any solution of system (1.2) to (1.7) satisfies

W, 0 < CE(O, @(I + t)-P

for p > 2.

Pro05 As in Theorem 3.1 we arrive at the following inequality:

d g(O) dt NE(t) + E(t) + 2 J(t)

:=&I)

< -Ko u

lu,12 + hlVu,12 + 1612 dA + a(u, u) +g’+“P •I a2u . 52

7 / :=N(t)

Since the energy is bounded, Lemma 4.2 implies that

w> 2 CJW (l+U-~)PMl--~)p 8 1+1/p 9 q azu 2 c Ig I a2u](1+(1-r)P)/(1-r)P .

It is not difficult to see that we can take N large enough such that C satisfies

c (E(t, u)} < L(t, u) < Cl {N(t) + gl+‘lp cl a2U}(1-r)p”1+(1-~)p).

From here it follows that

d --L(t, u) < --qL(t, U)o+(l-r)p)l(l-r)p,

(43

which implies that

at, u> G cao, 4 1

(1 + t)(‘-‘)P l


From this it follows that the energies decay to zero uniformly. Using Lemma 4.2 with r = 0 we get that

N(t) > cJv(t)(‘+P)‘~, g •I a2u 2 c {g 0 a2u}~1+p)~p.

Repeating the same reasoning as above we get

1 L(t, u) < CC(0, u)-

(1 + ty’

From this our result follows. The proof is now complete. cl

Acknowledgments

The authors express their appreciation to the referee for a valuable suggestion which improved this paper. The research was supported by a grant from CNP, (Brazil).

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Decay rates of solutions to thermoviscoelastic plates with memory