Existence et comportement asymptotique des solutions d'une équation de viscoélasticité non linéaire de type hyperbolique

Chapter 4

Exponential Growth

Abstract

_f

⁰⁰ g(s)ds < p -- 2

p -- ₁, by

0

Our goal in this chapter is to prove that when the initial energy is negative and p > m, then, the

solution with the L^u--norm g

rows as an exponential function provided that

using carefully the arguments of the method used in [16], with necessary modification imposed by the nature of our problem.

4.1 Growth result

Our result reads as follows.

Theorem 4.1.1 Suppose that m > 2 and m < p < oo, if n = 1,2, m < p < 2 (n -- 1) if n > 3.

n -- 2 --

p -- 2

00

holds. Then the unique local solution of problem

Assume further that E(0) < 0 and f g(s)ds <

0 p --1

(P) grows exponentially.

Proof. We set

H(t) = --E(t). (4.1)

By multiplying the first equations in (P) by --ut, integrating over Q and using Lemma 2.1.3, we obtain

t

₈

<

:

d

~ dt

_{0 1 9}

_Z =

2 kutk2

1 2 + 1 _@1 ~ 2 + 1

g(s)ds _A kruk2 ₂ (g ~ ru) (t) ~ p b kukp p

2 ;

0

1

1

= a IlutIC -- 2 (g' _o Vu)(t) + ₂ g(t) 11V ug + w rout g .

(4.2)

By the definition of H(t), (4.2) rewritten as

1

H'(t) = a Mutr_m -- 2 2 (g^' o Vu) (t) + ¹ g (t) 11V 74²₂ + w 1out1²₂ > 0, Vt > 0. (4.3)

Consequently, E(0) < 0, we have

1 2 + b

H(0) = ~₂ ku1k2 ₂ ~ 2 1 kru0k² _pMuce_p > 0. (4.4)

It's clear that by (4.1), we have

H(0) < H(t), Vt > 0. (4.5)

Using (G2) , to get H(t) -- b p One implies		2 0 1 3 Zt kukp 41 p = ~ 2 kutk2 2 + 1 @1 ~ g(s)ds A kruk2 2 + 2 1 (g ~ ru) (t) 5 2 0 < 0, Vt > 0. (4.6)		0 < H(0) < H(t) < b p
Iul^Pp . (4.7)

Let us define the functional

L(t) = H(t) + E./ n

ii ii 2 utudx + E 2 w IIVu112 . (4.8)

for E small to be chosen later.

By taking the time derivative of (4.8) , we obtain

L^'(t) = H^'(t) + E./ n

uutt (t, x)dx + F E 1lutg + Ew I SI

VutVudx

= [wIlVutg + a IlutI^m_m - ₂ (g^' o Vu) (t) + ¹₂g(t) 11V ug]

+E 1lutg + Ew I n Using the first equations in (P), to obtain

VutVudx + E I n

uttudx. (4.9)

Iuuttdx = bllurp HIV ug - col ~

VutVudx - a f n

1ut1^m-2 utudx

+I n

Vu

t I 0

g(t - s)Vu(s, x)dsdx. (4.10)

Inserting (4.10) into (4.9) to get

1

L^'(t) = wIlVutg + a MutE_r^m_i -₂ (g^' o Vu) (t) + ¹ 2g(t)1Vuk22

+ E Iutl22 --E 1Vuk^22 + E

t I 0

g(t - s) f Vu.Vu(s)dxds

+ EbIlurp - Ea I n

lutr^m-2 utudx. (4.11)

By using (G2) , the last equality takes the form

L^'(t) > w 1Vutk²₂ + a Iutr^m_m + E Iutk²₂ -E 1Vuk²₂ +EbIlur_p

(4.12)

+E

t I 0

g(t - s) f Vu.Vu(s)dxds - Ea f ~ n

lutr^m-2 utudo^-.

To estimate the last term in the right-hand side of (4.12) , we use the following Young's inequality

r XY < r

Xr + 8-q q

Y ^q, X, Y > 0, (4.13)

for all 8 > 0 be chosen later, 1 r So we have

+

1 q

m = 1, with r = m and q = m _ 1.

I

lutr^n-2 utudx <

I n

lutr^n-1 lul dx

MuM: + (mm 1) 8(m71) IlutIC , Vt > 0. (4.14)

8^m

<

m

Therefore, the estimate (4.12) takes the form

L^'(t) > w 11Vut11²2+ a Ilutr,,+E Mut11²2 ^-E 11Vu11²2+EbIlurp

t

_I

0

+ E

g(t - s) I Vu.Vu(s)dxds

E8^m a m

Murn^i Ea (m m 1) 8(mi^ni) 11u4117,

> w Ivutk22 + a IlutIC + E Iutk22 -- E 1Vuk²₂ + Eb 1uk1p

+ E r uk^22		t I 0	g(s)ds + E	t I 0	g(t - s) I Vu (t) [Vu (s) - Vu (t)] dxds
8 Ea^ln m	NC Ea (m m1) 8(^mm) IlutIC . (4.15)

Using Cauchy-Schwarz and Young's inequalities to obtain

L^' (t) > w 11V utg + a ( 1 - E ( ^{7 1} ₇₇^-₁¹) gmln1)) IlutIC + E 1lutg

E 1Vuk22 + Eb bun + E r uk^22

t I 0

g(s)ds

t

_I

0

E

Mull:

g(t - s) 11V u112 11V u(s) - Vu _(t)112 ds -- Ea^rl

m

> w 11V utg + a ( 1 -E (m m1) gm^ln1)) 11utr_nⁱ + E Ilutg + Eb Ng

(4.16)

₀

+ E @

1 2

t I 0

)g(s)ds -1 11V ug - E2 (g o V u(t) - Ea: Mull: .

Using assumptions to substitute for b Ilur_p . Hence, (4.16) becomes

L^'(t) > w IlVut11₂ + a ( 1 -- E (M m1) 8( min1)) Mud: + EllUt112

-FE (311(t) #177; 2 Ilutg + 2 (g 0 Vu) (t) + ^P₂ 1 -- I g(s)ds 11Vug

0t

1

2

-FE

t

_I

0

)

m

g(s)ds --1 11Vug -- E₂ (g o Vu(t) --ga_m Mull: .

> w 11Vutg + a ( 1 --E (m m -- ₁₎ 8( 7/7-)11 )) 11utr,,+ E (1 + 2) 1lUtg

(4.17)

+EallIVug + Ea2(9 0 Vu(t) -- Ea^ri Mur_rⁿ_i + €pH(t).

m

(1 --

f g(s)ds #177; ( P

where al = ) > 0, a2 = P 1 > 0.

0

2 p ) 22 2

In order to undervalue L^'(t) with terms of E(t) and since p > m, we have from the embedding L^P (Q) c-- L^m (Q) ,

m

Mur_rⁿ_i < C Ilugⁿ, < C (Ilur_p) P , Vt > 0. (4.18)

for some positive constant C depending on Q only. Since 0 < m P

< 1, we use the algebraic inequality

Zk < (Z #177; 1) < ( 1 #177; 1 ) (Z #177; W) , V Z > 0, 0 < k <1, w > 0, w

to find		m		(Mull;)
^P < K (Ng + 11(0)) , Vt > 0, (4.19)

1

0)

where K = 1 + H( > 0, then by (4.7) we have

IluEⁿ_i < C ( 1 + b) Mug, Vt > 0. (4.20)

P

Inserting (4.20) into (4.17), to get

L^'(t) > w IlVutg + a ( 1 -- E (m m - ₁₎ 8( ⁷¹⁷¹⁾) IlUtIr_m^l + E (1 + 2) 1lUtg

(4.21)

-Ec1 1Vuk22 + Ea2(9 ⁰ Vu(t) -- EC1114¹³_p#177; €pH(t).

where Ci = aC r771 ( 1 + b) > 0.

p

By using (4.1) and by the same statements as in [16], we have

2H(t) = - Iutl22 -^Ivuk22 +

Zt 0

g(s)ds kruk2 2 ~ (g ~ ru) (t) + 2b p

Ilull^Pp

(4.22)

~ ~Iutk²₂ IVuk²₂ -- (g Vu) (t) + 2pb Mur_p , Vt > 0.

Adding and substituting the value 2a3H(t) from (4.21), and choosing 8 small enough such that

a3 < min {al, a2} , we obtain

L^'(t) > w _Iloutll2 + a (1 -- E (m m1) 8 ( mln1)) Ilutrrni

+ E 11 + 2 -- a3) Iutk²₂ + E (al a3)1Vuk²₂

+ E (a2 -- a3) (g _o Vu(t) + E (2p ^b a3 -- C1) 1uk¹_p

+ E (p -- 2a3) H(t). (4.23)
Now, once 8 is fixed, we can choose E small enough such that

1 -- E (m m1) g^mlni) > 0, and L(0) > 0. (4.24)

Therefore, (4.23) takes the form

L^'(t) > E0 {H(t) + Ilutg+ I1Vug+ (g o Vu(t)) + Mur_p} , (4.25)

for some 0 > 0.

Now, using (G2), Young's and Poincare's inequalities in (4.8) to get

L(t) < 01 {H(t) + _11741122 + IlVu112 2},(4.26) for some 01 > 0. Since, H(t) > 0, we have from (4.1)

t

1

2 ^mutg 2 -10 1- f g(s)ds) 11V 2

-- (g _o Vu) (t)+bIIuIIP > 0, Vt > 0. (4.27)

0

Then,

1 2

0 @1 --

Zt 0

g(s)ds) Iloull2 < p b Murp

b
p

<

Mur_p + (g _o Vu) (t). (4.28)

In the other hand, using (G1) , to get

1 1

2 (1 -- l)11V ug 2

< (1 t

-- I g(s)ds) 11V ug

o

b

<

p

11u11^p_p + (g 0 Vu) (t). (4.29)

Consequently,

2b 11Vug < p

11u11^pp + 2 (g 0 Vu)(t) + 2l 11Vu11^22 , b,l > 0. (4.30)

Inserting (4.30) into (4.26) , to see that there exists a positive constant A such that

L(t) < A { H(t)+ 11ut11²₂+11Vu11²₂+ (g 0 Vu)(t)+ bp 11ur_p} , Vt > 0. (4.31)
From inequalities (4.25) and (4.31) we obtain the differential inequality

L'(t)

> it, for some ,u > 0, Vt > 0. (4.32)
L(t)

Integration of (4.32) , between 0 and t gives us

L(t) > L(0) exp (itt) , Vt > 0, (4.33)

From (4.8) and for E small enough, we have

By (4.33) and (4.34) , we have

L(t) < H(t) < b p

11u11^Pp . (4.34)

11u11^p_p > C exp (ut), C > 0, Vt > 0. (4.35)

Therefore, we conclude that the solution in the L^P--norm growths exponentially.