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

Chapter 2

Local Existence

Abstract

Our goal in this chapter is to study the local existence ( local well-possedness) of the problem (P), for u in C ([0, T] ,H¹ ₀(1)). For this purpose we consider, first the related problem for u fixed in C ([0,T] ,H¹ ₀(1))

₈

<>>>>>>>>>>> >

>>>>>>>>>>>>:

vtt - Lvt - Lv + f 0 t g(t - s)Lv(s, x)ds

+ jvtj^m~2 vt = juj^p~2 u, x 2 1, t > 0

, (2.1)

v (0,x) = u0 (x), vt (0,x) = u1 (x), x 2 1

v (t,x) = 0, x 2 [', t > 0

and we will prove the local existence of this problem by using the Faedo-Galerkin method. Then, by using the well-known contraction mapping theorem, we can show the local existence of (P). Our techniques of proof follows carefully the techniques due to Georgiev and Todorova [14], with necessary modifications imposed by the nature of our problem. The first step of our proof is the choice of the space where the local solution exists. The minimal requirement for this space is that u(t, x) be time continuous. The weak space satisfying the above requirement is C ([0, T] , H), where H = H¹ ₀(1) x H¹ ₀(1) is the natural energy space for (P).

2.1 Local Existence Result

In order to prove our local existence results, let us introduce the following space

YT =

{

u : u E C ([0, 7^1] , 11(1(Q)) , ut E C([0,7] ,11(1^-(Q)) n L^m([0, T] x Q)

}

· (2.2)

Our main result in this chapter reads as follows:

Theorem 2.1.1 Let (uo, ui) E (11j(Q))² be given. Suppose that m > 2, p > 2 be such that

--

max {m, p} < 2 (n -- 2 1) ' n > 3. (2.3)

Then, under the conditions (C1) and (G2), the problem (P) has a unique local solution u(t, x) E YT, for T small enough.

The proof of theorem 2.1.1 will be established through several lemmas. The presence of the term IuI^P-2 u in the right hand side of our problem (P) , gives us negative values of the energy. For this purpose we fixed u E C ([0, T] , 1/(1^-(Q)) in the right hand side of (P) and we will prove that our problem (2.1) , admits a solution.

Lemma 2.1.1 ([14], Theorem 2.1) Let (uo, 74) E (1/(1^-(Q))², assume that m > 2, p > 2 and (2.3) holds. Then, under the conditions (C1) and (G2), there exists a unique weak solution v E YT to the problem (2.1), for any u E C ([0, T] ,11(1^-(Q)) given.

The proof of the above Lemma follows the techniques due to Lions [26], in order to deal with the convergence of the non linear terms in our problem, we must take first our initial data (uo, u1) in a high regularity (that is, uo E H²(Q) n 1/(1^-(Q), and ui E 1/(1^-(Q) n L^2(m-1)(Q)).

Lemma 2.1.2 ([26], Theorem 3.1) Let u E C ([0, T] ,11j(Q)). Suppose that

uo E H²(Q) n HO(Q), (2.4)

ui E H^I₍(Q) n L^2(m-1)(Q), (2.5)

assume further that m > 2, p > 2. Then, under the conditions (G1) and (G2) there exists a unique solution v of the problem (2.1) such that

v E L°° ([0, T] , H²(Q) n 1/(1^-(Q)) , (2.6)

vt 2 L^°° ([0, T] , HO(Q)) , (2.7)

vtt 2 L^°° ([0, ⁷¹], L2(Q)) , (2.8)

vt 2 L^m" ([0, T] _x (Q)) - (2.9)

The following technical Lemma will play an important role in the sequel.

Lemma 2.1.3 For any v 2 C¹ (0, T, H²(Q)) we have

I i

st 0

dt[

t

1 d 1 d

g(t -- s)Av(s).71(t)dsdx = dt (g o Vv) (t) 2 1 ^g(s) I 1Vv(t)1² dxds

2

n

0

2 1 (g^' o VV) (t) + 2 g(t) 1 I n			1Vv(t)1^2 dxds.
where (g o u)(t) =	t I 0	g(t -- s ) 1 1u(s) -- u(t)1^2 dxds.

The proof of this result is given in [31], for the reader's convenience we repeat the steps here. Proof. It's not hard to see

I i st 0	g(t -- s)Av(s).71(t)dsdx = --	t I 0	g(t -- s) .1 V 7; (t).Vv(s)dxds
= -		t I 0	g(t -- s) I Ve(t).[Vv(s) -- V v(t)] dxds

t

I

0

g(t -- s) .1 Ve(t).Vv(t)dxds.

Consequently,

I .^ti st 0

g(t -- s)Av(s).vV)dsdx = 21

t I 0

g(t -- s) ddt I n

1V v(s) -- Vv(t)1^2 dxds

t

I

0

g(s) dt 2

n

d 1 1 1V v(t) 1² dx) ds

which implies,

I Zt

~ 0

2 1 d

g(t - s).6,v(s).71(t)dsdx = 2 I g(t - s) I 1V v(s) - Vv(t)1² dads

dt

0 ~

?

1 2

d dt

2 Zt 4 0

₃

_Z

g(s) jrv(t)j² dxds ₅

~

?

1 2

Zt 0

g'(t -- s) I

1Vv(s) - Vv(t)1^2 dads

+

1 gm I ~

1Vv(t)1^2 dads.

8 <> > >>:

~~wj= AjWi wi = 0 on['

j = 1, ..., in Q. (2.10)

This completes the proof. Proof of Lemma 2.1.2. Existence:

Our main tool is the Faedo-Galerkin's method, which consist to construct approximations of the solutions, then we obtain a prior estimates necessary to guarantee the convergence of approximations. Our proof is organized as follows. In the first step, we define an approach problem in bounded dimension space V_n which having unique solution v_n and in the second step we derive the various a priori estimates. In the third step we will pass to the limit of the approximations by using the compactness of some embedding in the Sobolev spaces.

1. Approach solution:

Let V = 1/(1^-(Q) n H²(Q) the separable Hilbert space. Then there exists a family of subspaces MI such that

i) c (dimV_n < 0o), Vn 2

ii) V_n -p V, such that, there exist a dense subspace in V and for all v 2 V, we can get sequence {vn}nEN 2 Vn, and v_n --p v in V.

iii) V_n C Vn+1 and UnEN*Vn = HO(2) n H2(l).

For every n > 1, let V_n = Span {wi,
·
·
·,wn} , where {wi}, 1 < i < n, is the orthogonal complete system of eigenfunctions of --A such that = 1, wi 2 H²(1) n L^2(m-1)(1) for all j = 1, n. Denote by {Ai} the related eigenvalues, where wi are solutions of the following initial boundary value problem

According to (iii) , we can choose vno, vni E [wi,
·
·., wn] such that

vno ~ _Xn ainwi --! uo in 1/₍1^-(Q) n H²(S), (2.11)

j=1

vn1 ~ _Xn Oinwi --! ui in 1/(¹)^-(Q) n L^2(m-1)(Q), (2.12)

j=1

solves the problem

₈

<>>>>>>>>>>>>>>> >

>>>>>>>>>>>>>>>>:

where

ajn =f~uowidx

Oi_n -- R ~uiwidx.

We seek n functions cp^riⁱ, _nE C² [0, , such that

v_n(t) = _Xn cp7(t)wi(x), (2.13)

j=1

f ((v^'_r(t) -- Av^'_n(t) -- Av_n(t))ndx

~

+ f f g(t -- s)Av_n(s)ds + 1v^'_n(t)1^n-2 (t)) ndx

; (2.14)

~ 0

=f ~	1u(t)1^P-2 u(t)ndx

v_n (0) = v_no, v_n^' (0) = v_ni

where the prime "'" denotes the derivative with respect to t. For every' E v_n and t > 0. Taking ~ = wj, in (2.14) yields the following Cauchy problem for a ordinary differential equation with unknown cp^riⁱ:

(,⁰7(⁰) = f uowi, (,⁰3^'ri(⁰) = f u1wj; = 1, ..., n

~ ~

_(pr_im (_t) +viin(t) + (4(0 + A

_~~m2 _'0n

+ ~~'0h j (t) j (t) = i(t)

₈

<>>>>>>>> >

>>>>>>>>>:

g(t -- s)(p^riⁱ(s)ds

_Rt

0

; (2.15)

for all j, where

3(t) = I

1U(t)1^19-2 U(t)Wi E C [0, 1] .

By using the Caratheodory theorem for an ordinary differential equation, we deduce that, the above Cauchy problem yields a unique global solution cp^riⁱ 2 H³ [0, T] , and by using the embedding H^m [0, T] y C^m-1 [0, T], we deduce that the solution (p^r_.; 2 C² [0, T]. In turn, this gives a unique v_n, defined by (2.13) and satisfying (2.14).

2. The a priori estimates:

The next estimate prove that the energy of the problem (2.1) is bounded and by using a result in [⁹], we conclude that_; the maximal time t_n, of existence of (2.15) can be extended to T.

The first a priori estimate:

Substituting n = v^'_n(t) into (2.14), we obtain

I	Zv^00 n(t)v^0 n(t)dx ~ ~			en(t)en(t)dx -- f ~		Avn(t)en(t)dx
+f ~		Zt 0	g(t -- s)Avn(s)dsen(t)dx + f ~		I n(t)1^m-2 (t)en(t)dX (2.16)
= I ~	f(u)en(t)dx.

for every n > 1, where f(u) = lu(t)1^P-2 u(t), Since the following mapping

L²(Q) L²(Q)

uIUI^P-2 u

is continues, we deduce that,

juI^P-2 u 2 ([0, T] , L²(Q)) .

+

1 2

d dt

2 Zt 4 0

So, f 2 11¹ ([0, T] , H¹(Q)). Consequently, by using the Lemma 2.1.3 and (G1) we get easily 1 d2 dt Ilvn,(0112 + 11V vn(t)112 + 2dt 11V vn(t)11

_{3 0 1}

_{Z Z}t

1 d

g(t ~ s) jrv_n(s) ~ rv_n(t)j² dxds 5 ~ _@krv_n(t)k² g(s):ds _A

0

2

2 dt

~

~	1 2		Zt 0	g'(t -- s) I	vn(s) -- Vvn(t)1^2 dxds +^12g(t)1Vvn(t)k^22 +1en(t)kmm
= I ~		f(u).en(t)dx:

Therefore, we obtain

d
dt

En(t) -- f f(u):v⁰ _n(t)dx + krv⁰ n(t)k2 ₂ + kv⁰ n(t)km m

1

= ₂(g^' o Vv_n)(t) -- 1 g (t) 11V v_n(t)g ,

where

₈

<

1

E_n(t) = 2 :

0 ¹Me_n(t)11₂ + 1 -- I 9(s)ds 11V v_n(t)g + (g 0 V v _n) MI ,

0t

is the functional energy associated to the problem (2.1). It's clear by using (G2), that

dt nE (t) -- I n

f (u).v^'n(t)dx + 11V en(t)112 + Ilvn(t)E^ni < 0, Vt > 0.

Which implies, by using Young's inequality, for all 8 > 0

d
dt

E_n(t)+11Ve_n(t)11₂+11e_n(t)k: < ₄¹₆. Ilf(u)g +6^.11e_n(t)11²₂. (2.17)

Integrating (2.17) over [0, t] , (t < T), we obtain

En(t) +				t I 0	11V en(s)11^22 ds +			t I 0	11 vn (s) kC ds
<	1 48	T I 0	11 f(u) g ds + 8			T I 0	Men(s)112 ds + 2 (Iluing + 11Vuong)

Since f 2 H¹(0, T, I(1^-(Q)), we deduce

En(t) +

t I 0

11V en(s)11^22 ds +

t I 0

Men(s)kri ds < CT (Mu ing + 11V u °mg) , (2.18)

for, 8 small enough and every n > 1, where CT > 0 is positive constant independent of Ti. Then, by the definition of E_n(t) and by using (2.11) , (2.12), we get

(

o

t

117/_n(0g + 1 -- I s(s)ds 11V v_n(t)g + (g 0 V v_n)(t) KT, (2.19)

t

I 0

and

Men(s)km7 ds < KT, (2.20)

and

_Zt

0

1Vv⁰_n(s)1²₂ ds < KT, (2.21)

where KT = CT (11u1n11²2 + uOn11²2) , by (2.19), we get t_o = T, Vn.

However, the insufficient regularity of the nonlinear operator, lvtl^m-2 vt, with the presence of the viscoelastic term and strong damping, we must prove in the next a prior estimates that, the family of approximations v_n defined in (2.13) is compact in the strong topology and by using compactness of the embedding H¹([0, T] , H¹(Q)) y L²([0, T] , L²(Q)), we can extract a subsequence of v_n denoted also by v, such that v^', converges strongly in L²([0, T] , L²(Q)). To do this, it's suffices to prove

''

that vn is bounded in L^°([0, T] , Hj(Q)) and v_n is bounded in L'([0, T] , L²(Q)), then by using Aubin-Lions Lemma, our conclusion holds.

The second a priori estimate:

Substituting n = wi in (2.14) and taking -Awl = Ajwi, multiplying by wi'ⁿ(t) and summing up the product result with respect to j, we get by Green's formula.

Z

vvn^».vvrcdx + I ~

Aen.Aendx + f ~

Avn.Avn^' dx - I ~

Zt 0

g(t - s)Avn.Av^0ndsdx

aaxi n -2 _n^{\ a}_a^v:ⁿ_i dx + ^E./ ^(17/1m )

j=1 ~

n

(2.22)

=	4(m - 1) m2			En I (axi ^a oencv env dx. i=1 ~
= I ~	Vf(u).Vv^0ndx.

As in [26], we have

En I ° 1^{m-2 avn'}

dx

j Oxi ⁿ Oxi

i=1 ~

= (m - 1)

En , 2 I a 1m2 2^tjvn) dx Oxi n axi (2.23) i=1 ~

Also, the fourth term in the left hand side of (2.22) can be written as follows

I Zt

st 0

g(t -- s)Av_n.Ae_ndsdx = -- 21 g(t) _Ilovnll2 + ²¹ (g^' o Av_n)

~

1 2

d dt

8 < :

(g o Avn) -- 1Avnk^22

Zt 0

g(s)ds

9 = ;

. (2.24)

Therefore, (2.22) becomes

_{2 0 1 3}

1 d

2 dt

_Zt

0

₄krv⁰ nk2 ₂ + (g ~ ~v_n) + k~v_nk² _@1 ~ g(s)ds _{A 5}

2

r (1v/711 2 v' )

aXi m 2 2

n) dx (2.25)

t=i

~

4(m -- 1)

_m2

+

+ 1164112 + ₂g (t) Ilovnll2 -- ₂(g^' 0 Av_n)

^-1 ~

Vf(u).Vv^0ndx.

Let us define the energy term

K_n(t) = 2 [11Ve_n11₂ + (g _o Av_n) + 1lAv_ng (1 -- I g(s)ds)1 (2.26)

Then, it's clear that (2.25) takes the form

dtKn(t) -- I

V f (u).Vv^'ndx + IlAv^'n11^22

2

(lenl m₂ -2 e_n)) dx

#177; En I (axi

i=1

(2.27)

= ~	2 (g (t) 11 Av ng 2 ) + (g^' 0 Avn).

Using (G1) , (G2) and integrating (2.27) over [0, t] , we obtain

Km(t) +

Xn i=i

Zt 0

I

t (aaxi (lenl m2 2 en)^2 dxds + I 0

16v^0n1^22 ds

~

_Z

_ZT

0

vf(u).vv⁰_ndxds + 21 (1Ivvin + IAvong). (2.28)

Obviously, by using Young's inequality, we get

ZT 0

Z

Vf(u).Vendxds < 4^16.

ZT 0

1Vf(u)1^2 2ds + ~

ZT 0

IovnI22 ds.

Inserting the above estimate into (2.28), to get

Kn(t) +

Xn i=i

Zt 0

Z

~ @ ~ 2 Zt m2 jv^0 nj 2 v0 dxds + n @xi 0

.6411^22 ds

< CT(1Vvink²2 + 11.Avong)
·

Thus,

Km(t) +

Xn i=i

Zt 0

Z

~ @ ~ 2 Zt m2 jv^0 nj 2 v0 dxds + n @xi 0

1.4k^22 ds < CT,

for, 8 small enough and every n > 1, where CT > 0 is positive constant independent of Ti. Therefore, this equivalent by the definition of K_n(t)

0

0 IlVe_n11₂ + (g _o Av_n) + 11Av_ng 1 -- I g(s)ds)< CT, (2.29)

and

and

Xn i=i

Zt 0

Z

( @~ 2

m2

jv⁰ nj 2 v0 dxds ~ CT ; (2.30)

n

_Zt

0

^axi

.641²₂ ds < CT. (2.31)

Then, from (2.29), (2.30) and (2.31) , we conclude

e_n is bounded in L^'([0, T] , 1^l((Q)), (2.32)

v_n is bounded in L^'([0, T ] , H²(Q)), (2.33)

^a (Iv l ^m-2 e_n) is bounded in L²([0, , L² (Q)) , i = 1, n. (2.34)

Oxi n 2

The third a priori estimate: It's clear that

d dt

2 Zt 4 0

0

t

1 g(t -- s)Av_n(s)ds = g(0)Av_n + I g(t -- s)Av_n(s)ds. (2.35)

Performing an integration by parts in (2.35) we find that

d dt

2 Zt 4 0

g(t -- s)Av_n(s)ds = g (t) _Aura) +

1I g(t -- s)Avas)ds. (2.36)

0 t

Now, returning to (2.14), differentiating throughout with respect to t, and using (2.36), we obtain

I	(erat) -- Av^00n(t) -- Aeri(t)) ndx

+I

n

0 Zt

@0

)g(t -- s)Ae_ri(s)ds + g(t)Au_rio + (m -- 1)(le_ri(t)r^-2 vat)) ndx

= I ~

(f(u))^')ndx, (2.37)

where, (f (u))^' = 0 _at f

. By substitution of n = v⁰⁰h(t) in (2.37), yields

I	vZ(t)vat)dx -- I Avat)4(t)dx -- I ~ n				Aer(t)4(t)dx
+I n		0 @	t I 0

)g(t -- s)Ae_ri(s)ds + g (t)Au_rio v⁰⁰_n(t)dx

+(m - 1) I le_n(t)1^m-2 v⁰⁰_n(t)v⁰⁰_n(t)dx (2.38)

=

I

(f(u))^' v⁰⁰_n(t)dx.

The fourth term in the left hand side of (2.38) can be analyzed as follows. It's clear that Lemma 2.1.3 implies

fi st 0	g(t -- s)Aeri(s).vat)dsdx = --1 n	t I 0	g(t -- s)Ver(s).Vv^00n(t)dsdx
1 d 1 d = dt (g ~ rv^0 n) (t) ~ 2 2 dt				t I 0	g(s) 1Ver(t)1^22 ds

(2.39)

1 1

₂ (g o Ve_r) (t) + ₂g(t) 1Ve_r(t)1²₂ :

Also,

_Z

(m ~ 1)hjv⁰ n(t)jm~2 v⁰⁰ n(t); v⁰⁰ _n(t)i = 4(m ~ 1)

_m2

~

~ @ ~ 2

m2

jv⁰ nj 2 v0 _n(t) dx: (2.40)

@t

Inserting (2.39) and (2.40) into (2.38), we obtain 8 < 1 d d dt kv^00 nk2 2 + krv^00 nk2 2 + 1 dt d krv^0 nk2 2 + 1 2 2 2 dt :				(g o Vert) (t) -- IlVen11^22		Zt 0	g(s)ds	9 = ;
+	4(m -- 1)f m2 ~		(at^a (17/1 m2 2 en(t) ))^2 dx + (g (t)Auno) I		v^00n(t)dx (2.41)
= I ~		(f(u))^' v^00n(t)dx + 2^1 (g^' o Ven) (t) -- 12g(t) IVvnI22 :

Let us denote by

8 <^11^1 n(t) _-- 21 :

1174112 + + (g 0 Vv^'n) (t) + (1 --

Zt 0

g(s)ds)11Vv^'n11^22

9 = ;

: (2.42)

2

(at ^a (17/1 m₂ 2 e_n(t) )) dxds

4(m ~ 1) ~_n(t) +

_Zt _Z

0

_m2

+1 0	krv^00 nk2 2 ds + (g(0)~un0)		Zt 0		Z	v^00n(s)dxds
1 2 + 1 ~ 2 kv00 n0k^2 2 + 2 1 krvn1k2 4		ZT 0		~ ~(f(u))0~~2 2 ds + ~			ZT 0	11411^22 ds.

We obtain, from (2.41) , (G2) that

dtT^n(t) I	Z (f(u))^' .vat)dx +^4(rn ~1) m2 ~		2 (at^a (17/1 m2 2 en(t) J) dx
Z+ krv^00 nk2 2 + (g(0)~un0) ~		v00 n(t)dx

1 1

= 2 (g⁰ ~ rv⁰ _n) (t) ~ ₂g(t) krv⁰ n(t)k2 2

< 0.

Integration the above estimate over [0_; t], we conclude

Then, for 8 small enough, we deduce

-- ^11^1n(t) + 4(m 2 1) m		t I 0	I	( a ( len 1^m-2 2 VI ) ) 2 dxds at n
t +I 0	1Vv^00n1^22 ds + (g(0)Auno)				t I 0	I	v^00ndxds (2.43)

< CT (114011²2 + 11Vvnig) .

In order to estimate the term Ilv_n^"011²₂, taking t = 0 in (2.14), we find

11Vn0112 = I n	Avni.vn^ttodx + I II		Avno.vnttodx
-I		vni ·vnd ttox + Ivnl 1 m-2 I		f(u(0)).v^"nodx.

n

Thanks to Cauchy-Schwartz inequality (Lemma 1.2.10), we write

1V^00n01^22 < 1V^00n012	IlAvn1112 + IlAvn0112 + Ilf(u(0))112 + (I17)1n12(m-1) dx ~			2 1)	;
which implies, by using (2.11) and (2.12) , that 1140112 < IlAvni112 + IlAvn0112 + Ilf(u(0))112 + (I1v1n1^2(m-1) dx ~		1 2	< C.

(2.44)

Then,

WI/_n(t) < LT, (2.45)

t

_I

0

I

4(m -- 1)

_m2

( _:t ( IVInl m-2

Vfn) ) 2 dxds < LT, (2.46)

and

t

_I

0

and

t

_I

0

1VV¹⁰_nk²₂ ds < LT, (2.47)

I v⁰⁰_ncXdS < LT. (2.48)

Chapter 2. Local Existence			30
where LT > 0. From (2.11), (2.12) and (2.44) - (2.48), we deduce v00 n is bounded in L^°°([0, T] , L^2(1)), v is bounded in L^°°([0,T] ,H^1 0(1)), @ m-2	1,	..., m.	(2.49) (2.50) (2.51) (2.52) (2.53) (2.54) (2.55)
jv^0 2 is bounded in L^2([0, T] L^2(1)), i v0 = nj n , @t ( ~
3.Pass to the limit:
By the first, the second and the third estimates, we obtain v is bounded in L^°°([0, T] , H^2(1) fl H^1 0(1)), v0 n is bounded in L^°°([0,T] ,H^1 0(1)), v00 n is bounded in L^°°([0, T] , L^2(1)), v0 n is bounded in L^m((0, T) x 1).

Therefore, up to a subsequence, and by using the (Theorem 1.1.5), we observe that there exists a subsequence v_T of v and a function v that we my pass to the limit in (2.14), we obtain a weak solution v of (2.1) with the above regularity

vT -~ v in L^°°([0, T] , H¹ ₀(1) fl H² (1)),
v0 ~ -~ v^ein L^°°([0, T] , H¹ 0(1) fl L^m(1)),

v00 ~

-* v^,,in L^°°([0, T ] , L^2(1)).

By using the fact that

L^°°([0, T] , L²(1)) ,! L²([0, T] , L²(1)),

L^°°([0,T] ,H¹ 0(1)) ,! L²([0,T] ,H¹ 0(1)).

We get

v0 n is bounded in L²([0, T] , H¹ ₀(1)),

v00 n is bounded in L²([0, T] , L²(1)),

therefore,

v0 n is bounded in H¹([0, T] , H¹(1)). (2.56)

Consequently, since the embedding

H¹([0,T] ,H¹(1)) ,! L²([0,T] ,L2(1))

is compact, then we can extract a subsequence v,^' such that

71, --> V^' in L²([0,T] , L²(Q)). (2.57)

which implies

v0 ~--> V

^0 a.e on (0, T) x ~.

By (2.20), we have

v₇,^' is bounded in L^m([0, T] x Q).

and by using Theorem 1.2.2

1,141m-2 ~ *n in L^ml([0, T] , (Q))

m where m -- 1

= m^'.

The estimates (2.34) and (2.46) imply that

m-2

174 2 _VT^I --
· v in H¹([0,T] , H¹(Q)

by using the fact that, the mapping u ^m-2 u is continuous, (Lemma 1.2.9) and since the weak

topology is separate, we deduce

'9 =lelm-2 rvr

(2.58)

m-2

~ = jv^0j

2 v^f

(2.59)

Then, by using the uniqueness of limit, we deduce

1741m-2 VT * vr in Lml([0, ^77], p^rl(c)) (2.60)

1741

m-2 2 vr ~ *7)11

m-2 ^2 V^I in H^1([0, 1^7] , H^1(Q) (2.61)

Now, we will pass to the limit in (2.14), by the same techniques as in [26]. Taking n = wi, n = T and fixed j < T,

I v,⁰⁰(t).widx + I V 7), (t).V wi dx + I Ve_T(t).Vwidx
~ ~

- I ~ 0		g(t -- s)V 7), (s).V widsdx + I ~	Iv^'T(t)I^m-2 vT(t).widx (2.62)
= I ~	f(u).widx.

We obtain, by using the property of continuous of the operator in the distributions space

I v00,(t).widx *~ _Z v^"(t).widx, in D^' (0, T)

~

_Z Vv_T(t).Vwidx * _Z Vv(t).Vwidx, in L^°° (0, T)

~

I VeT(t).Vwidx *~ _Z Ve(t).Vwidx, in L^°° (0, T)

~

I Zt ~ 0	Zg(t - s)Vv,(s).Vwidsdx *~ ~			Zt 0	g(t - s)Vv(s).Vwidsdx, in L^°° (0, T)
Z		1e,(t)1^m-2 (t).widx *~	Z ~	1v^'(t)1^n-2 e(t).widx, in L^' (0,T)

We deduce from (2.62), that

_Z vⁿ(t).widx + I

~

Vv(t).Vwidx + f Vv(t).Vwidx

~

- I st 0		g(t - s)Vv(s).Vwidsdx + I ~	1v^'(t)1^n-2 v^'(t).widx (2.63)
= I ~	f(u).widx.

Since, the basis wi (j = 1, ...) is dense in Hj(a) n H² (a) , we can generalize (2.63) , as follows

I		v^"(t).(pdx + I ~		Vv(t).V cpdx + I ~	Vv^'(t).Vcdx
- I .^t1 12 0			g(t - s)Vv(s).V cpdsdx + I ~			1V^1(t)1^m-2 Vf(t).(pdx
Then,	= I ~	f(u).cpdx, Vcp E 1(1^-(a) n H^2 (a) .

v E ([0,7],H²(a) n 1j(a)) ,

vt E ([0,T] ; H¹₀(a)) ;

vtt E ([0,7],L²(a)) ,

vt E L^m ([0,7¹] x (a)) : This complete the our proof of existence.

Uniqueness:

Let v1, v2 two solutions of (2.1), and let w = v1 -- v2 satisfying :

00 w

-- Aw -- Aw^' +

t I 0

g(t -- s)Awds + (lyr^-2 711 -- 14r-2 v2) = 0. (2.64)

Multiplying (2.64), by w' and integrating over Q, we get

1 d

2 dt

⁺I

g,

0t(Ilw^'_ri(t)11₂ + 1 -- I g(s)ds) 11V w_ri(t)g + (g o V w_ri)(t))

(1 _vt_i lm-2 _vt₁ -- 141m-2 v2) w^tdx

= -- IlVw^'_ri(t)11²₂ + ²¹ (g^' o Vw_ri)(t) -- 12 g(t) 11 Vw,,,(t) g

:

Denote by

0 J(t) = Ilw^'_ri(t)11²₂ + 1 -- I g(s)ds)11Vw,,,(t)g + (g 0 V w_ri)(t). (2.65)

0t

Since the function y i-- lyr^n-2 y is increasing, we have

I (le_i lm-2 v1 -- 141m-2 v2) w'dx > 0

and since

¹₂(g^' 0 V7_n)(t) < 0,

we deduce

(d _j ( _{t\ 0}) .

(2.66)

dt ^{k 1} )

This implies that J(t) is uniformly bounded by J(0) and is decreasing in t, since w(0) = 0, we obtain w = 0 and v1 = v2.

Proof of Lemma 2.1.1.

As in [14], since D(Q) = H²(1), we approximate, uo, u1 by sequences (uo) , (u_ni) in D(Q), and u by a sequence (0) in C ([0, T] , D(Q)), for the problem (2.1). Lemma 2.1.2 guarantees the existence of a sequence of unique solutions (0) satisfying (2.6) -- (2.9) . Now, to complete the proof of Lemma 2.1.1, we proceed to show that the sequence (0) is Cauchy in YT equipped with the norm

kuk2 YT = kuk2 _H + kutk2 L^m([0,7]x1) '

where

MuM^2H = o X <T

{I [74 + l Mull (x, t)dx ; ~

Denote w = vⁱ¹ -- v^A for ,u, given. Then w is a solution of the Cauchy problem:

8 <>>>>>>>>>>> > >>>>>>>>>>>>:

wtt - Aw - Awt + L(w) + k(vr) - k(v~t) = f(u^~) -- f(u^~), x 2 Q, t > 0 w(0, x) = upo - uo,^wt(0, x) = uu1 - uo, x 2 Q w(t,x) = 0, x 2 I', t > 0

, (2.67)

where,

k(vt) = Ivr I^m-2 vr
f(u^~) = Iu^lI^P-2 u^li

L(w) =

t I 0

g(t -- s)Aw(s, x)ds.

1

2

d

₈

<

:

dt

The energy equality reads as

_{0 1 9}

Zt =

0

kwtk2 ₂ + _@1 ~ g(s)ds A krwk2 ₂ + (g ~ rw)(t) ;

+I

n

~ ~

k(v~ _t ) ~ k(v~ _t ) wtdx + IVwtk²₂

(2.68)

= I ~			(f(uP) - f(u^~)) wtdx + 2^1 (g^' o Vw) (s)ds -- 2^1 IVw(s)k^22			t I 0	g(s)ds.
~~~~~~	I	(f(u) -- f(u)) (v -- v) dx		~~~~~~	< C(Iu11 + MuL)^1-2 Mu - uMH 1v - vkH ,

The term,

I	~ ~ Z ~ ~ ~ ~~~v ~~~ ~ m2 k(v~ t ) ~ k(v~ t ) wtdx = jv~ t jm~2 v t ~ v v t ~ v dx t t t

is nonnegative.

We need to estimate

fulfilled for u, u, v, v 2 1/(1^-(Q), where C is a constant depending on Q, l, p only. Then, Holder's

inequality yields, for 1 q

+

1 n

+

1 2

2n =^1 (q = n - 2),

~~~~~~

I

(f(u^~) - f(u^~)) wtdx

~~~~~~

=

~~~~~~

I

( ~ ju~jp~2 u ~ ~~u~^p2 u ~ v t ~ v dx t

~~~~~~

< C Me -- u9 _L9 4ⁱ -- vi L₂ (1101V_-2) + 11u91:-(P2 _₂₎ J.

(2.69)

The Sobolev embedding Lq c- 1/(1^-(Q) gives

Mu^~ - u9i9(n) < C MVO - Vu9L2_(n).

Then,

1101113n-02 ₃_₂) + 11u11¹⁹:₀₃_₂₎ < C(11011¹³L2g1) +W 11¹⁹,;(²_n)).

The necessity to estimate Iluⁱll_n(_p_₂) by the energy norm Mul_H requires a restriction on p. Namely,

we need n(p - 2) < 2n then the Sobolev embedding L^q c- 1/(1^-(Q) gives n - 2,

ku

^Ik^p~2 n09-2) < 11u111-1². Therefore, (2.69) takes the form

~~~~~~

I

(f(uP) -- f(u^~)) wtdx

~~~~~~

~ ~ ~ ~ ~

~ C _~v

~ t ~ _v ~ ~

^t ~L2() _~ru^~ ~ ru~~ ~L2(~) kru~kp~2

L2() + ~ru~~ ~p2 (2.70)

L2(~)

under the fact that

t

I

0 (g^' o Vw) (s)ds <0,

we conclude

~		t I 0	(g^' o Vw) (s)ds + (g 0 Vw) (t) + 1Vw(t)k^22			t I 0	g(s)ds > 0.
Thus,	2 2 2 11w(t, .)11H <-- 2 11w(0, .)11H + C			t I 0	MVu^~ - our ~L2(n) Mwt(s, .)1H ds.

The Gronwell Lemma and Young's inequality guarantee that

Ilw (t , .)11 H < Ilw (0 , .)11 _H + CT Me- _u91C([07],H) .

Since

P^I*, .) -- v(t, .) 11_H < C_!Iv^u (0, .) -- v (0, .)11_H + CT 110 -- u9!I!IC([027],H) , (2.71)

then fv^~I is a Cauchy sequence in C([0, t] , H), since ful and {v(0, .)} are Cauchy sequences in C([0, T] , H) and H, respectively.

Now, we shall prove that {v^~_t } is a Cauchy sequence, in L^m ([0, T] x Q), to control the norm kv~ t k2 L^m([0_;T ]x~) . By the following algebraic inequality

(a1a1^{m-2 --}0101^m-2) (a -- 0)> C la -- 01^m, (2.72)

which holds for any real a, 0 and c, we get I (^k(4) - k(4)) wtdx = I

(vr Ivr^n-2 - 74 vi^m-2 (vr - 4) dx

2

< C^,14⁴ - 74 .

LM ([0,7] x11)

This estimate combined with (2.68) gives

_{!I !I}

!I4 - 74 11² __,r,

t

I

0

+CR

!IL ([0,t] x12) < C IIv^A(0, .) - v(0, .)11L,T,([0,t]xSZ)

!I!Iu~ - uIILm([0,t]xf2) P^I*, .) - v(t, .)11 L^m([0,t] x12) ds.

So by using Cromwell Lemma, we obtain {vn is a Cauchy sequence, in L^m ([0, T] x Q) and hence fv^~I is a Cauchy sequence in YT. Let v its limit in YT and by Lemma 2.1.2, v is a weak solution of (2.1).

Now, we are ready to show the local existence of the problem (P) Proof of Theorem 2.1.1.

Let (uo,u1) 2 (H¹₀(Q))² , and

R² = (1Vuok22 + Ilui g)

For any T > 0, consider

MT = fu 2 YT : u(0) = up, ut(0) = ui and Mull_yT < RI .

Let

0: MT--MT

u i-- v = 0(u).

We will prove as in [13] that,

(i) ⁰(MT) g MT.

(ii) 0 is contraction in MT.

Beginning by the first assertion. By Lemma 2.1.1, for any u 2 MT we may define v = 0(u), the unique solution of problem (2.1). We claim that, for a suitable T > 0, 0 is contractive map satisfying

0(MT) _c MT.

1

2

₈

<

:

Let u 2 MT, the corresponding solution v = 0(u) satisfies for all t 2 [0, T] the energy identity :

_{0 1 9}

Zt =

kv0(t)k2 ₂ + _@1 ~ g(s)ds A krv(t)k2 ₂ + (g ~ rv)(t) ;

0

t +f 0

1Vv^'(s)1^22 ds +

t I 0

1v^'(s)1C)ds

(2.73)

1 ~kv1k2 = 2 + krv0k2 ~ + 2 2

t I 0

I

lu(s)1^19-2 u(s)v^'(t)dxds.

We get

1 2 1 2 2 ^Ilv(t)IlyT 2 Ilv(^0)11yT +

t I 0

I

lu(s)1^19-2 u(s)v^'(t)dxds. (2.74)

We estimate the last term in the right-hand side in (2.74) as follows: thanks to Holder's, Young's inequalities, we have

I 1u(s)j^1-2 u(s)v^'(t)dx < C 1uKT _1v117 ,

then,

Ilv(t)g,T < Ilv(0)117, + CR^P

t I 0

IvIIT ds,

where C depending only on T, R. Recalling that uo, ui converge, then

Ilv(t)Il_yT <11v(0)11₁₇₇, +CR^PT.

Choosing T sufficiently small, we getlIvIl_yT < R, which shows that

0(MT) C MT.

Now, we prove that 0 is contraction in MT. Taking wi and w2 in MT, subtracting the two equations in (2.1), for v1 = 4(wi) and v2 = 0(w2), and setting v = v1 -- v2, we obtain for all n 2 Hj(Q) and a.e. t 2 [0, T]

I

vtt.ndx + I n

VvVi7dx + f n

Vvtviidx + f n

t I 0

g(t -- s)VvVndsdx

+I

n

(Ivtl^m-2 vt) ndx

= I ~

(Iwil^p-2 w1 -- 1w21^P-2 wOndx. (2.75)

Therefore, by taking n = vt in (2.75) and using the same techniques as above, we obtain

Ilv(t, .)g,, < C

t I 0

~kw1k^p~2 YT + 11w^21G^2) 11w 1 -- w^2llyT Ilv (s , .)11iTT ds . (2.76)

It's easy to see that

2 2 2

11⁷*, .Ali = 11⁽¹⁾(w¹) - (1)(w2)1117t a Ilwi -- w²llyt , (2.77)

for some 0 < a < 1 where a = 2CTR^P-2.

Finally by the contraction mapping theorem together with (2.77), we obtain that there exists a unique weak solution u of u = 0(u) and as 0(u) 2 YT we have u 2 YT. So there exists a unique weak solution u to our problem (P) defined on [0, T], The main statement of Theorem 2.1.1 is proved.

Remark 2.1.1 Let us mention that in our problem (P) the existence of the term source ( f( u) = luI^P-2 u ) in the right hand forces us to use the contraction mapping theorem. Since we assume a little restriction on the initial data. To this end, let us mention again that our result holds by the well depth method, by choosing the initial data satisfying a more restrictions.