Несуществование решения затухающей системы нелинейных волновых уравнений типа Кирхгофа

1. Introduction and Previous Work

Let us consider the problem

(|ui|^{m 2}ui)0 + jj |D^Kux|2dx^ (—A)^Kui + (a|ui|^k + b|u2|^l) u i = fi(ui,u2);

Ωγ

(1.1)

(|u^,₂|^m-2u^,₂)0 + j j |D^KU2|²dx^ (-A)^KU2 + (c|u2|⁶ + d|ui|%) u2 = f2(Ui,U2),

Ω where all terms must be alive, we will prove that the solutions of (1.1) cannot exist for t> 0 with positive initial energy, where ui(x, 0) = uio(x) E Hq(D), i = 1, 2,                           (1.2)

ui(x, 0)) = u_ii(x) E L^m(D), i = 1, 2,

(1-3)

(1-4)

(1-5)

and boundary conditions

∂ju

—— = 0, x e dD, i = 1, 2, j = 0,1, 2,..., к - 1, ∂νj where v is the outward normal to the boundary.

In the present paper, we study the system (1.1), with fi(ui,U2) = (p + 1) |^ai|ui + U2|(p i)(ui + U2) + bi|ui|(p- )ui|u21)] ,

(p-3)          (p+1)

f2(ui,u2) = (p + 1) ai |ui + U2© )(ui + U2) + bi|u2| 2 U2|ui | 2    ,

and the paramelres ai > 0. bi > 0. p > 3. 7 > 0. m > 2. k,l,9, %, к > 1 satisfying p > max (m — 1, k + 1, l + 1, 9 + 1, % + 1, 2y + 1) .                    (1.6)

In (1.1). u i = u i (t,x). i = 1, 2, where x E Q is a boundei 1 domain of Rⁿ (n > 1) with a smooth boundary dQ, t >  0 and a, b, c, d are nonnegative constants.

We mention here that

|DKu|2 = (AK/2u)2 for par value of к and

|DKu|2 = |V(A(K-1)/2u)|2 for odd к, where

n          2

IVu|² = Ё(lu) • Au = X dn.

^ V dxi )^ dx^

i=1

This kind of systems appears in the models of nonlinear Kirchhoff-type. It is a generalization of a model introduced by Kirchhoff [11] in the case n = 1; this type of problem describes a small amplitude vibration of an elastic string. The original equation is:

phu tt + Tu t =

Eh

P0 + 2L

L           \

У |u x (x,t)|² ds I u xx + f, 0

(1.7)

where 0 6 x 6 L arid t >  0. u(x,t) is Ilie lateral displacemenI, al, the space coordinate x arid the time t. p the mass density, h the cross-section area. L the length. Po the initial axial tension, t the resistance modulus, E the Young modulus and f the external force (for example the action of gravity).

The blow up of the gender of our problems in the single equation has been considered in [18]; it was established a blow-up result for certain solutions with positive initial energy. In [14] local existence and blow up of the solutions, of the same equation have been studied.

A related problems with к = 1 have attracted a great deal of attention in the last decades, and many results have been appeared on the existence and long time behavior of solutions. For the literature we quote essentially the results of [2-5], [7], [10-12], [15, 17, 19, 20, 22, 23, 30] and references therein.

The systems of nonlinear wave equations (1.1) go back to Reed [24] who proposed a similar system in three space dimensions but in the absence of the viscoelastic and damping terms. This type of system was completely analysed; for example, in [2], the authors studied the following system:

I

u tt - Au + |u t |^{m 1} u t = fi(u,v), v tt - Av + |v t |^r-1v t = f2(u, v),

(1.8)

in Q x (0,T) with initial and boundary conditions and I,lie nonlinear functions f1 aiid /2 satisfying appropriate conditions and in the case where a = b = c = d = y = 0, m = 2, к = 1. They proved under some restrictions on the parameters and the initial data many results on the existence of a weak solution. They also showed that any weak solution with negative initial energy blows up in finite time using the same techniques as in [8].

In the work [19], the authors considered the nonlinear viscoelastic system:

'                           t utt — Au + J g(t - s)Au(x,s) ds + |ut|m-1ut = f1(u, v),

<               0                                                        x E Q, t > 0,       (1.9)

vtt - Av + / h(t - s)Av(x, s) ds + |vt|r—1 vt = f2(u, v), where

( f₁(u, v) = a|u + v|^2(p+1) (u + v) + b|u|^p u|v|^(p+2),

(1.10)

[ f2(u, v) = a|u + v|2(p+1) (u + v) + b|u|(p+2) |v|pv, and they prove a global nonexistence theorem for certain solutions with positive initial energy, the main tool of the proof is a method used in [25].

In the case of y = 0, к = 1, m = 2, problem (1.1) has been studied recently in [22] focusing on the global well-posedness of the system of nonlinear wave equations utt - Au + (d|u|k + e|v|l) |ut|m 1ut = fi(u,v), jutt - Av + (d0|v|6 + e0|u|p) |vt|r-1vt = f2(u, v),

(1.И)

in a bounded domain Q C Rⁿ, n = 1, 2, 3, 0 < r, m < 1, with Dirichlet boundary conditions. The nonlinearities f₁(u, v) and f₂(u, v) act as a strong source in the system. Under some restriction on the parameters in the system, they obtain several results on the existence and uniqueness of solutions. In addition, they prove that weak solutions blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms. This last result was extended by A. Benaissa, Ouchenane, and Zennir in [3] with positive initial energy, r, m > 0 and for n >  0.

Our main theorem addresses to generalize earlier results in the literature. We will improve the influence of a strong sources with positive initial energy, which lead to blow up of solutions for all t >  0 in Theorem 3.1.

(p + 1)F (u1 ,u2 ) = [u1f1(u1 ,u2 ) + u2 f2(u1,u2 )] =

^ a1|u1 + u2|^p+1 + 2b1 |u1u2| ² ) ] ,

(1.12)

where

∂F            ∂F я— = f1(u1,u2), я— = f2(u1,u2).

du1               du2

(A2) There exist a positive constant c₁ = 2^pa + b such that

F(u1 ,u2) 6 C1 X |ui |^p+1.

i =1

(1.13)

(1.14)

We introduce the following definition of weak solution to (1.1)—(1.4).

Definition 2.1. A pair of functions (u₁,u₂) is said to be a weak solution of (1.1)—(1.4) on [0,T] if u1,u2 G Cw([0,T],HK(Q)), u1,u2 G Cw([0,T],L^m(Q)), (u_w,u2o) G H K (Q) x H^Q)

(u11,u21) G Lm(Q) x Lm(Q) and (u1,u2) satisfies.

tt

J j(\U₁\^m-2U₁Yфdxds + j kD^Kuik2^Y J D^кU1D^кфdxds

0Ω                0        Ω

t

+ j J ((a |ui| k

+ b |u₂|l) и1Ф dxds =

t j j f1(u1 ,u2)фdxds;

0Ω

t

(|u2|

0Ω

t m-2u2)0^ dxds + j ||DKu2k2Y j DKu2DK^ dxds

(1.15)

Ω

t

+ Я^

+ d |u11 %) u'₂^ dxds =

t j^ j f2(u1 ,u2)^ dxds

0Ω

for all test functions ф, ^ G H0K(^) И L^m (fi). for alrilost all t G [0, T]• AVhere C_w ([0,T],X) denotes the space of weakly continuous functions from [0, T ] into Banach space X.

The energy functional E(t) associated to our system is given by:

E(t) = mml X H IK + _2(Y+I) X |D^Ku , ll2(^Y+1) - j F(U1,U2) dx.      (1.16)

The following Sobolev-Poincare inequality will be used frequently without H0K(fi) C Lp(fi), for mention

(1-17)

{ 1 < p,          if n = к, 2k,

1

if n > 

3k.

n -2 κ

We first state (without proof, it is similar to that in [23]) a. local existence theorem for n = 1, 2, 3. Unfortunately, due to thc strong nonlinearities on fi. f₂ the well known techniques of constructing approximations by the Faedo-Galerkin allowed us to prove the local existence result only for n 6 3.

Theorem 2.2. Let n = 1, 2, 3. Suppose that (1.17) holds. Then, there exists a local weak solution in the sense of Definition 2.1 of problem (1.1)—(1.4) defined on [0, T] for some T >  0, and (u1,u₂) satisfies the energy inequality

E ^{(t) +}

t

/(/ (^|U1^{(T)|k + |U}2^(T)|1 ^ (ul) s Ω

2dx

(1.18)

+

Q

(|u 2 (t )|⁶ + |ui (t )|%) (u2)² dx

dT 6 E (s)

for all T > t > s >  0. where E(t) is given in (1.16).

3. Results

Our main results read as follows

Theorem 3.1. Suppose that (1.6), (1.17) hold. Then any solution of the problem (1.1)(1.4), with initial data satisfying

^X |D^Ku_i0k2 >a2 , i =1

(1.19)

and

E «U i ik m + or 7 kD^KU i 0«2^(Y+1) -    F (uio,V2o) dx <  d

\         2(Y + 1)

i=1                                             Q blows up for all time, where the constants a1 and d are dehned in (1.21).

We introduce the following:

(1.20)

B = n p+1 ,

p+1

a₁ = B ^1-p,  d =

-

2(y + 1) p + 1

a2,

(1.21)

where n ^ the constant in (1.28).

Lemma 3.2. Suppose that (1.17) holds. Let (u₁ ,u₂ ) be a solution of (1.1)—(1.4). Assume further that

X «D - U i ok! >a2 , i=1

(1.22)

and

m

-

i =1

m

- «U i ik m + 7W—TnkD U i 0«₂⁺ ) - [ F (uio ,U2o) dx < d.

2(Y + 1)

Q

(1.23)

Then there exists a constant a₂ >  a₁ such that

X IID^Ku i k2 > ^a2, i =1

(1.24)

and

1/(p+1)

( (p + 1) / F (u₁,u₂)dx)        >  Ba₂ (V t >  0).

Q

(1.25)

<1 By the definition of energy functional, we have

E(t) =  • Xi№ '2.1 X D -  F (U1-U2) dx i=1                (Y ) i=1                       Q m - 1 V'' II 0||m I 1      V'' II К ||2(y+1)       1      ||     ,     ||P+1 , oil       llpl-

= ^—m~ /. hui^m ⁺ 2(7+1) ^^ D ui^2     - P"+T [hu^{1 +} u²^P+1 ^{+ 2}II^U1 u²^ P+1

By using Minkowski’s inequality and embedding H^fi M- L(p+1) (Q), we get

/ ₂          \ p+1      / ₂                  \ p+1

i =1                        i =1

Holder’s and Young’s inequalities give us p+1

p +1     /                      \ P^{+1          2}                   \ ²

ig^Hф 6 (|Ы!р+1 ku2kp+1) 2 6 c YllD^illfo1)     .       (1.27)

\ i=1                 /

Then there exist n > 0 such that p+1

,            p+1         ²_         w A ²

ku + U2kp+1 +2kuiU2kp +1 6 n YilDWiT )     .           (1.28)

• 2\i=1

By definition of B wo gel,

E(t) > ЖУЙ X D' "2lY+1) — + »“1 + U2^P+1 + 2Ь “2" p+1 ,=1                                                                 2

p+1 2                    / 2\ ”

• >    y+n X ID*",«П11 - +г X D ^

(Y + ) i=1                  p +    i=1/ p+1

• >    y+i) X HD^2 ⁺ - Bl¹^ (x ilDM+S ² = f (a),

(' + ) i=1                    1 +      i=1/ where a2 = P2=1 |DKui|2l7+1)- We can verify that the function f is increasing for 0 < a < ay decree sing for a > ay f (a) ^ —to as a ^ +to. and

|p+1)

f (a!) = ------- a 2 --aP⁺¹ = d,                     (1.30 )

f^{( 1)}     2(y + 1) ¹ p + 1    ¹         ’                           '      '

where a ₁ given in (1.21). Therefore, since E (0) <  d. there exists a₂ > a₁ such that f (a₂) =

^E<0):                                  ..2i.+„

Now we set a2 = P,=1 ||DKUioЦ2 7   , then by (1.29), we have f (ag) 6 E(0), which implies that a 0 > a2. To establish (1.24), we sup pose by contradiction that P2=1 |DKui(to) k2(Y+1) < a2 for some tg > 0 to choose tg such that P2=1 |DKui(to)k2lY+1) > a 1-

Again using of (1.29) leads to

E (to) > fl E kD^Ku,(to)k2^lY+1) ) >  f (a 2) = E(0).

,=1

This is impossible since E (t) 6 E(0) (V t E [0,T)).

To prove (1.25), we exploit the definition of E, to get

p+1

<1 Proof of Theorem 3.1. We set

H (t) = d - E (t).                                   (1.31)

By using (1.16), (1.31) we get

H0(t) = f (к« + U(t)| l ) K1(t)|² dx

(1.32)

+ j ( |u2(t)|⁶ + |ui(t)| % ) |u⁰₂(t)|² dx >  0 (V t >  0).

Therefore,

0

-         XIID^KU112^(Y+1)+ — 11 u + U2 11p+1 +2 11U1U2 11

2(Y + 1) i^"      ²P + 1 L ^{1    2}p^{+1      1 2}

From (1.24), we obtain that for all t > 0 the estimates hold d--7-------7      11DKU (to) 112(Y+1) +--11 Ui + U2 11 p+1 + 2 11U1U2 11 (

2(y + 1) ^{2 2 iv} 2           p + 1 ii^{1      2}11p+1      ii ^{1 2}11 (p^¹)

(P+1)

-

2^ a2 + p^[ 11U1 + U211 P+1 +211U1U211 ф

1                 1 Г

< -7+f а1 + Tvy 11 u1 + u211 P+1+ 211 uru211 (p+i) p + 1      p + 1 |_2

(p+1)

.     11U1 + U2 11 P+1 +2 11U1U2 11 S p + 1

Hence by (A2), we have

⁰<^H(0) 6 ^H(t) 6 -+1 XIIui IIP+1. i=1

Then we introduce

L(t) = H^1-CT(t) + e / ^Uiluil^m-2uidx,                      (1.33)

Q i=1

ε p — (k + 1) p — (l + 1)

0 < a 6 min p—(----), p—(----)■ p+1 p+1

p — (% + 1) p — (9 + 1) (p — (m — 1)) p + 1 p + 1     m(p + 1)

(1.34)

We will show that L(t) satisfies

L0(t) > ^L^1+v(t),    for all t > 0, v > 0, £ > 0,

(1.35)

defined in [0, to). By taking a derivative of (1.33) and using (1.1), we obtain

2                  2

L(t) = (i - »)h--№0(t) +to Hie+e£ ^D^Km^2^(Y+1) i=i            i=i

-ε

j ui (|ui^(t)|k + ^|u2^(t)|l)

Q

ui dx — e

j^u2 (^|u2^(t)|^ ⁺ |u1⁽t) |%) Q

u⁰₂dx

+e

j (uif1(ui ,U2 )+ U2 f2(ui,U2)) Q dx.

Then

L0(t) = (1 — a)H'(t)H⁰(t) + eX HE + eX D^K- 2 ' i=1            i=1

—e j ui (|ui(t)|^k + |u2(t)|l) u⁰1 dx — e j U2 (|u2(t)|⁶ + |ui(t)|%) u'₂dx

(p+1) \

^+e ( ||^{ui +} u² ||p+i +2^^uiu2^ (p+1) ) . \                                     2  /

By exploiting (1.16) and (1.21), equation (1.36) takes the form

L0(t) > (1 — ,)H-(t)H0(t) + e^{m + 2(Y + 1)(}m — ¹⁾

m

+e2(Y + 1)H(t) — e2(Y + 1)d + e2 X||D^Ku, i=i

i=i

2(Y+i) ik2

ku⁰ik^mm

—e j ui (|ui(t)|^k + |u2(t)|l^ uidx — e j U2 (|u2(t)|^e + |ui(t)|%^ u'₂dx

^{+e(1 —} (_pY-+₁ )) (l^{|ui +} u²l^|p+i

(p+1) \

^{+ 2|}uiu2^|(p+1)   .

We will estimate, for some Constance Ai, A2 > 0, two terms as j (|ui(t)|k + |u2(t)I1 ) |uiui|dx

Q j (|ui(t)|k + |u2(t)I1} | ui|2dx, Q

(1.36)

6 Ai У (|ui(t)|^k + |u₂(_t)|l) |ui|²dx + 4A_

Q and j (|u2(t)|6 + |ui(t)|%) |u2u2|dx

Q

(1.37)

6 A2 j (|u2(t)|⁶ + |ui(t)|%) |u21²dx + 4^ j (|U2(t)|⁸ + |ui(t)|%) |u2|²dx. Q                                Q

Then,

L0(t) > (1 - ,)H-’(t)H0(t) + £m + (Y + )(m - П X ku'km mm i=1

+£2 X ^D^KUi^2^(Y+1) +²(Y + W(t) i=i

• ^+ef1Y I 1 Y (llul ⁺ u2 Hp+l ^{+ 2 ||u}1^u2 II p+1 Y \ p + 1               ^p+1            — у

  (1.38)

  
  

• -eAi j (|U1 (t)|^k + |u2(t)|^l) |U1 |²dx - £4^1 j (|U1 (t)|^k + |u2(t)|^l) |u11²dx

• -e\2 j (|u2(t)|⁶ + |U1 (t)|%) Ы2dx - £4^2 j (|u2(t)|⁶ + |u1(t)|%) |u21²dx.

Consequently, by using Young’s inequality for some 5, 01 > 0, we have j (|U1(t)|k + |u2 (t)|l) |u1|2dx = 11U111 k+2 + j |u2|l |U1|2 dx

Q                                      Q k+2 + —— d(l+2)/l 11П9 1 |l+2 + — d-(l+2)/(2) 11 nJ 11+2

• ⁶    u¹k+2 ⁺l + 2° u²1+2 ⁺l + 2 °           u¹1+2’

and j (|u2(t)|e + |U1(t)|%) |U2|2dx = 11U2 11 0+2 + j |U1|%|U2|2dx

Q                                      Q

⁰⁺²-и % x⁽%^+2)/%      %⁺²I ² x^-(%^+2)/(2)      %⁺²

• ^{6    11} u2 ¹¹0+2 ⁺ %+^ °1       ¹¹ u1¹¹%+2 ⁺ %+^ °1          ¹¹ u2 ¹¹%+2 •

Then,

L'(t) > (1 - ,)H^-"(t)H'(t) + £^{m + (Y + )(m}- 1) X ku'km m^m i=1

+£2 X kD^KUik2^(Y+1) + 2(y + 1)£H(t)

i=1

• ^+£f1--^Y . 1 ^ ) f^ku1 ^{+ U}2k^p+1 ^{+ 2}1¹ u1u2 1¹p+1 )

^{p + 1}                              2

• ^{- £}4X1 / (^|u2^{(t)|0 + |u}1^(t)|%) ^|u2^|2d^{x -£}4X2 / 0u1^(t)|k^{+ |u}2^(t)|1) ^|u1^|2dx QQ

(11    11⁰⁺²I % X⁽%^+2)/% 11      %⁺²I ² X^-(%^+2)/(2)11    11 %⁺²J

• ^{- £}A1 ¹¹ u2 ¹¹0+2 ⁺ %+^ °1       ¹¹ u1¹¹%+2 ⁺ %+^ °1          ¹¹ u2 ¹¹%+2j

• - £A9 (11 П1 11 k+2 + —— d(l+²)/l 11     ¹⁺² +       d-(l+²)/(²) 11 и 1 11 l⁺²^

• ^{£ X2}    u¹k+2 ⁺l + 2 °        u²1+2 ⁺l + 2 °           u¹1+2 ‘

Choosing A₁. X₂ such that

—= m₁H ^a(t),  -7T-= m₂H ^a(t), m₁,m₂> 0.

(1.39)

4X1                 4X2

Using (1.39) and the fact that

H y^j

Q

(|ui(t)|k + |u2(t) I¹) |U1 (t)|²dx +

/

(I “2(t)I⁶ + |ui(t)|%) |u2(t)|²dx

(Vt > 0),

Q

l,o obtain for M = m₁+ m₂and assiimption (A2).

+e2 X ^^ки_гH2^(y+1) + 2(Y + 1)eH(t) + ec2 X ||uikP+1

i=1                                           i=1

_ ¹„<7 0+2 % .⁽%⁺²⁾/% %^{+2 2} ,-^-(%⁺²⁾/⁽²⁾%⁺²^ -^E4m?^{H (t) H}“^{2 H}^+2 ⁺ %+2 ^1 H^U1H%+2 ⁺ %+^⁵1 ^H“² H%+2^ -e_±_H(t) (Hu H^ + _l_5^(1+2)/1 Hu₂Hl+2 + -2-d^-(1+2)/(2) Hu Hl+2) .

4m₂^{1 k+2}l + 2 ^{2 1+2}l + 2 ^{1 1+2}

Since (1.6) holds, we obtain by using condition (1.34)

тт_СТ/,\ц lli+2 / fll ||O"(p+1) + (i+2)    11 цст(р+1)ц _Mi+2\

^{H (t)|u1} ki+2 ⁶ c³ ^u¹ ll(p+1)        ^{+ H}u^2H(p+1) ku¹ ki+2y ;

(1.40)

rr_o-/.\II l|j+2 /    (II ^H-(p+1)+(j+2) . II ^Ha(p+1)^H||j+2\

(H'(t^U Н^+2 6 c4 ^Ы^и)       + ||u1 H(p\1)J Hu2Hj+2j , where i = k,l, % and j = 9, %,l. Then

L0(t) > ((1 - ,) - Me)H-’(t)H⁰(t) + Em ^{+ (Y + )(m}- ¹⁾ X ku0km m^m i=1

+e2 £ kD^Ku,kl^lY+1) + 2(y + 1)eH(t) + EC2 £ ku,kp:1 i=1                                        i=1

• ¹    H     11a(p ■ 1)-Ю ■ 2) ^H     ||a|p ■ 1) H     ^H0+2

• ^-E4m1 c⁴ (^Hu2^H(p+1)        ^{+ H}u^1H(p+1) ^Hu^2H9+2)

_ ¹      % c(%+2)/%„ /      ^alp•1•^l%•^{2            P(}p- 1        %⁺2\

^-E4m;%+2⁵1      c³ l^Hu^1H(p+1)        ^{+ H}u2^H(P+1) ^Hu^1H%+2,)

• ^{1     2}_x-(%+2)/(2)           a(p+1) + (%+2)         -(p+1)      %+2

• ^-Erm;%+i⁵1        c⁴ (^Hu^^lp+1)       ^{+ Hu1} lh+1) ^H“^2H%+2/

^{1 H}    ||^CT(p+1) + (k+2) . ^{H    H}a(p+1)^H    ||k+2

^-E4m2c³ vu^{1 H}(p+1)        ^{+ H}u^2H(p+1) ^Hu^1Hk+2)

1l

-^E 4m₂ 7+2⁵     c4 (^H“2^HT1)⁾ (  ) ⁺ «“1 H

> Hu2 ^H

1₊2)

__e__2__d-(l+²)/(²)_c ; H_niH<^(p+1)+(i+2)₊^|lu^|г^(p+1) Hn₁H¹⁺²^

^E 4m₂l + 2 ⁵            c^{3 H H} u¹(p+1)         ^{+H H} u²(p+1)^{H H} u^{1 H H}1+2 ‘

By using (1.34) and the algebraic inequality zv 6 (z + 1) 6

1 +

a

(z + a)

(Vz > 0, 0 < v 6 1, a > 0),

(1.41)

we have. for all t > 0,

a(p+1)+j+2            (p+1)                       (p+1

llui||(p+1)       ⁶ b ^l^ui^(p+i) ⁺ H⁽⁰⁾) ⁶ b ^I^ui^(p+1) ⁺ H^(t)) ,

(1.42)

where b = 1 + 1/H(0). j = k, 6, l, % and i = 1, 2. so that we obtain

L0(t) > ((1 - ,) - Me)H-(t)H0(t) + £m±_h±_llm-h X ц^K mm i=1

+£2 X ||D^Ku,Ц^ + 2(y + 1)£H(t) + £C2 X ^Uihp+1

i=1                                              i=1

¹               (p+1)                   ^(p+1)     9+2

• ^-£ 4m1 c⁴ V 'u² p+ ⁺H^(t)) ^{+ flu1 fl}(p+1) llu² H9+2)

¹    % _x(%+2)/%             (p+1)                   ^(p+1)      %+2

• ^-£4m1 %+2⁰1      c³ v(ll^U1H(p+1)^{+ H(t)}) ⁺ Hu²H(p+1) H^u1H%+2)

• ^{1     2}   r—(%+2)/(2)              (p+1)                    ^(p+1)      %+2

• ^-£4m1 %+"2⁵1          c⁴ v Ulu^2fl(p+1)^+H(t)) ⁺ Hu^1fl(p+1) llu^2fl%+2)

• ¹               (p+1)                   ^(p+1)      k+2

• ^-£4m2c³ v    ^u- 'p" -^{+ H(t)})⁺ nu²H(p+1) iiu¹ Hk+2)

• -£     -l-d^(l+2)/lC4 (b (Ik l|(p+1) + H(t)) + flu fl;⁽p+1⁾ flu₂ Hl+2)

₄m_{2 l + 2}         ^{4        2} (p+1)                 ¹ (p+1)    ²l+2

__e__2_5—(l+²)/(²)_c(₆(||_nifl^(p+1)_{+ H}(_t\)₊ IL/., "'p ' 'JL/ ¹' ²)

• ^£ 4_m2 1 + 2⁰            c³ b 11 u¹ (p+1) ⁺ H^{(t) +fl}I u² (p+1) 11 u¹111+2 "

Also, since (X + Y)^s 6 C(X^s + Y^s), X, Y > 0, making use of (1.34) we conclude

(p+1)^-(^j+ 2) 11U2 11 flL) 11U111 j+2 6 ^    ⁽P⁺¹⁾

o-(p+1) (p+1)

(p+1)-(j+2)

6 |Q|    ⁽P⁺¹⁾

j+2 \ p+1

U2 L+1) I |U11L1

(1.43)

(p+1)-(j+2)

6 |Q|    ⁽P⁺¹⁾

'         q(p+1) + (j+2)              ^(p+1) + (j + 2) \ ^p+1

C011 U2||(p+1p⁺¹⁾     + C⁰⁰11U111 p+1 p⁺¹     )      6

c= xi 1 и* 11 ;p:i).

0 _ ^g(p+1)     _r00 — i⁺²           X _ L 7 „

+ ^{nele c}_CT(p+1)+(i+2) ' ^C CT(p+1)+(i+2) ' ^{1 1} j k^,l,%-Similarly,

W^(p+1)    j+2             (p+1)

(p+1) 11 u² 11 j+2 ^{6 C}6 2_^ 11 ui 11 (p+1),

(1.44)

for j = 6. %, l.

Taking into account (1.43), (1.44), we deduce

L0(t) > ((1 - a) - Me)H'(t)H0(t) + Em+jT+Jm—l) X ^ui^m i=1

+e2 X ||DKui^2(Y+) + 2(y + 1)EH(t) + Ec2 X llui HP+1 i=1

—e----c b (Hu2||⁽p⁺¹⁾ + H(t)) + ce X    ^|(p⁺¹⁾

4m 1 ^{4        2} (p+1)               ⁶ i (p+1)

• 1     % x(%+2)/%              (p+1)(

• ^—E4m%+2⁵1     c^{3 b}(^|u1|(p+1)^+H(t))⁺ c⁵^^||ui|(p+1)

• 1                    X

_ ^{1    2} £^-(%^+2)/(2)       (     ⁽P⁺¹⁾         A , „ ^{X (}P⁺¹⁾

• ^—E4m1 %+2^d1        c^{4 b ||U2|}(p+1)^{+H(t) +} c⁶2^ ^||ui|(p+1)

• —E-----C₃ | b f ||ui Ц/ _n + H(t)) + Cs     I IL I

_4m^{3         1} (p+1)                ^{5 i} (p+1)

—E            '       C₄ bf ¹1U₂¹1 (p+1) + H(t)) + C6 X^||U¹1 (p+1)

4m₂1 + 2         ^{4        2 (}p^{+1)               6}i ⁽p⁺¹⁾

___J-(l+2)/(2)     b (^{11    (p+1)}        Й +C.X ¹¹U •^{11 (P+1)}

^E₄m₂1_{+ 2}⁰           c^{3 b 11} u¹ (p+1) ⁺ H^{(t) +} c⁵^ ¹¹ ui¹¹ (p+1) ■

Then

L(t) > ((1 — a) — Me)H^-"(t)H0(t) + Em ^{+ (Y + )(}m - 1) X ku0km m^m i=1

+ e2 £ kD^KUi k²2^(Y+1) + e 2(y + 1) + —- C7 + —- C8 H(t)          (1.46)

4m14

i=1

^+e ( ^c2 ^{+     c}7 ^{+     c}9 ) X^kui^kP+1"

4m1     4m2

i=1

For large values of m1 aiid m₂ we can find positive constants A and B such that

L0(t) > ((1 — a) — Me)H^-ct(t)H0(t)

(1.47)

m + 2(y + 1)(m — 1) x 0 m            X

+ E          m           / , 11ui11m + EAH(t) + EB2^ 11ui p+1’ i=1

We pick e small enough so that ((1 — a) — Me) > 0 arid L (0) > 0. Consequently. I,here exists Г > 0 such that h

i=1

Thus. "■ have L(t) > L(0) > 0, for all t > 0. On the other hand, we have

1                                                                      \ i

L^1—ct (t) = (H ^1-ct (t) + e         uiluilm^-2ui(x₁t)dx'\

^V             Q i=1                    '

-σ

6 C10 H(t) +

/ ^ Ui|u0 |^m-1dx

Q i=1

1 —CT

(1-49)

By Holder’s and Young’s inegualities, taking (1.6) into accaunt, we estimate

1                                                                                 _z                                                    „

1 —ct           1        m—1                             ^m—m—

■•' dx 6 iiui iim-- Miim.- 6 Cinm-    (|Ы1р+т,,1—mCT' + н iim), i = 1, 2,

Q and also using (1.34), we have

/ £ Ui |ui|^m-1

Q i=

1 —CT

u1 u01 m-1

Q

1—

+

j U21U21 m^-1

Q

1 —CT

(1.60)

m

6 c(ei i и-11p+^—т+e11 ui 11m

By using again (1.34) and (1.41) we get

m

11 Ui ii3+5" 6 b( 11 Ui 11+ + H (t))   (i = 1,², vt > 0).                (1.51)

Therefore,

1 L1—ct

(t) 6 C11 [h(t) + X ku,k<p+1>i=1

+ E kui km i=1

(Vt > 0).

(1.52)

With (1.52) and (1.48), we arrive at

,                           1

L⁰(t) > a₀L ~ (t) (V t > 0).                            (1.53)

Finally, a simple integration of (1.53) gives the desired result. B

• 5.    Comments and Question

Remark. Let us mention that our main contributions in this article is the study of the influence of strong source terms on the existence of solutions with positive initial energy and in the higher-order function spaces, where f1, f₂ drive the solution of our system to blow up for all t if they dominate the damping terms, for large values of p.

Noting that one need carefully following the proofs of results in this paper to prove the nonexistence of solutions of the viscoelactic cases, using some well known assumptions on the memory terms, but it will be interresting to see the energy decay rate which will be according with that of the relaxation functions.

Question: One can consider the problem

'                                                                               t u1 - ф (kVuik2) Aui + ^ CkVuxk2) J gi(t - s)Aui(s) ds = 0,

0t                                          (1-54)

u2 - Ф (kVu2k2) Au2 + ^ (kVu2k2) J g2(t - s)Au2(s) ds = 0, and may ask questions on asymptotic behavior of the solutions (If it existes): as time goes to infinity, what is the asymptotic behavior of solutions? More generally, what is the long time behavior of solutions when initial data vary in any bounded set in a Sobolev space associated with the problem (1.54).

Acknowledgments. The author want to thank the referee for his/her careful reading of the proofs.