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Владикавказский математический журнал 201-5, Том 17, Выпуск 4, С. 44-58
ON THE ABSENCE OF SOLUTIONS TO DAMPED SYSTEM OF NONLINEAR WAVE EQUATIONS OF KIRCHHOFF-TYPE
Kh. Zennir, S. Zitouni
In higher-order function spaces, some techniques are used to give the nonexistence result to system of wave equations in the Kirchhoff type, to generalize earlier results in the literature.
Mathematics Subject Classification (2000): 35L05, 58J45.
Key words: blow up, Kirchhoff-type, wave equations, degenerately damped system, strong nonlinear source, positive initial energy, higher-order.
1. Introduction and Previous Work
Let us consider the problem
(KP-X/ + (f |DKui|2drV (-A)Km + (aim|k + 6M1 ) u[ = /i(m,U2);
<y (1.1)
(|u/2|m-2U/2)/ + (y |DKU2|2drJ (-A)KU2 + (c|U2|0 + d|ui|e) u2 = /2(Ui,U2),
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) G Hg(n), i = 1, 2, (1.2)
Ui(x, 0)) = ui1 (x) G Lm(n), i = 1, 2, (1.3)
and boundary conditions dj Ui
dv j
= 0, x G dfi, i = 1,2, j = 0,1,2,..., к - 1, (1.4)
where v is the outward normal to the boundary.
In the present paper, we study the system (1.1), with
/i(ui,u2) = (p+ 1) ai|ui + u2|(p_1)(«i + nz) + b^u^^-1 U\\U2\^~L
\ ( 11 (p-3) (p+l)
/2(ni,n2) = (i>+1) ai|ui + U2\(p~ ){ui + u2) + 6i|n2| 2 u2\ui\ 2
(1.5)
and the paramétrés a\ > 0 bi > 0 p > 3 7 ^ 0 m ^ 2 k,l,9,g,K ^ 1 satisfying
p > max (m - 1,k + 1, l + 1,0 + 1, g + 1,27 + 1). (1.6)
© 2015 Zennir Kh., Zitouni S.
In (1.1), u = ui(t,x), i = 1, 2, where x £ fi is a bounded domain of Rn (n ^ 1) with a smooth boundary dfi, t > ^d a, b C d are nonnegative constants. We mention here that
|DKu|2 = (Ak/2u)2 for pax value of k
and
|DKu|2 = |V(A(K-1)/2u)|2 for odd k,
where
This kind of systems appears in the models of nonlinear Kirchhoff-type. It is a generalization
n=1
a small amplitude vibration of an elastic string. The original equation is:
phuu + TUf = + J \ux(x,t)\2dsj uxx + f, (1.7)
where 0 ^ x ^ L and t > 0 u(x,t) is the lateral displacement at the space coordinate x and the time t, p the mass density, h the cross-section area, L the length, P0 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 k = 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:
iutt — Au + |ut|m-1 ut = fi(u,v), [vtt - Av + |vt|r-1vt = f2(u, v),
in fi x (0,T) with initial and boundary conditions and the nonlinear functions f1 and f2 satisfying appropriate conditions and in the case where a = b = c = d = 7 = 0 m = 2, k = 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 £ fi, t > 0, (1.9)
vtt — Av + / h(t — s)Av(x, s) ds + |vt|r-1vt = f2(u, v), 0
where
f1(u,v)= a|u + v|2(p+1) (u + v) + b|u|pu|v|(p+2), 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 7 = 0 k = 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|1) |ut|m-1ut = f1(u, v),
(1.11)
vtt — Av + (d'|v|e + e'|u|p) |vt|r-1vt = f2(u, v),
in a bounded domain fi 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,
r, m > 0 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 t > 0
2. Notations and Preliminaries
The constants Ci, i = 0,1,2,..., used throughout this paper are positive generic constants, which may be different in various occurrences. We take a = b = c = d = a1 = b1 = 1 for convenience.
(Al) There exists a C^function F : R2 ^ R such that
t +1 (p+i) i
{p+ l)F(ui,n2) = [ui/i(ui,u2) + u2/2(ui,u2)] = ai\ui+u2\p+ +26i|mn2| 2 ; (1.12) where
dF dF
^ = /i(m,u2), — =f2(Ul, u2). (1.13)
(A2) There exist a positive constant c1 = 2pa + b such that
2
F(u1 ,u2) < |ui|p+1. (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 £ Cw([0, T], Hq (fi)), ui,u'2 £ Cw([0,T],Lm(fi)) (uw,u2o) £ HK(fi) x HK(fi),
(«11, «21) G Lm(fi) x Lm(fi) and (ui , u2) satisfies, t t J J(luilm-2ui)'^dxds + J ||DKuiIl27 y DKuiDK^dxds
on on
t t
+ J J ^(a |ui|k + b |u2|'j u'^dxds = J J fi(ui,u2)0dxds;
on on
t t
J J(lu'2lm-2u'2)'^dxds + J ||DKu2ir | D^D^dxds
o n o n
t t
+ dlH') *** = //f2(ui .„
(1.15)
for all test functions ^ G Ho(fi) n Lm(ft), for almost all t G [0,T], Where Cw([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:
^ = / P{Ul,U2)dx. (1.16)
m i=i 2(7 +X) i=i n
The following Sobolev-Poincare inequality will be used frequently without mention H0K(fi) C Lp(fi), for
i 1 < p, if n = k, 2k,
(L17)
I if n>3/e.
We first state (without proof, it is similar to that in [23]) a local existence theorem for n = 1, 2, 3. Unfortunately, due to the 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 ^ 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 (ui,u2) satisfies the energy inequality
t
\k i |„, /„\|A / „,/ \2,
E(t) + J [J (|ui(t)|k + |u2(T)l') (ui)2dx
s n (1.18)
+ / (|u2(r)|0 + |ui(r)|e) (u2)2 dx^dr < E(s)
n
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 anv solution of the problem (1.1)-(1.4), with initial data satisfying
2
EHDKuio||2 >«2, (1-19)
i=1
and
2
Im i 1 ||
(1.20)
¿(lln.ll^^H^oll^) - f F(uw,v20)dx<d i=1 n
blows up for all time, where the constants a1 and d are defined in (1.21). We introduce the following:
1 P + l /1 1 \ 9 ,
B = 71^+1, ai=Bd=[—----a?, (1.21)
' V2(7+l) P+1/ 1 ^ J
where n is 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
£||DKuo||2 >a2, (1-22)
i=1
and
E^IKC + ^iyll^oll^)-/F(uw,u20)dx<d. (1.23)
i=1 n
Then there exists a constant a2 > a1 such that
2
]T||DKui||2 >a2, (1-24)
i=1
and
f r \ 1/(P+1)
l(p +1) / F(u1,u2)d^) ^ Ba2 (V t ^ 0). (1.25)
n
< By the definition of energy functional, we have
2
2
n/jA m — 1 V^ II / l|m
EV) = —— 2JWL +
1
m
2(y + 1)
2 i ^|DKui|2(7+1) — F (u1,u2) dx
m1
m
EM / l|m
IIu,-1 +
im
1
i=1
'2(7 + 1)
EID
M2(7+1)
u
2
P + 1
I I IIP+1 I nM
P1 + u2 IIp+1 + 2Iu1 u2
P+l 2
P+l 2
By using Minkowski's inequality and embedding H^fi ^ L(p+1) (fi), we get
< c(^ ||D u,j||2 .
p+l p+l n1+n2||!lU2£ii (¿Ik||0 2 2
lp+1
(1.26)
1
1
1
Holder's and Young's inequalities give us
p+i
p+i 2
2
< c E ||D
\i=1
||uiU2||^i < (||«l||p+i IM|p+i 2 v
Then there exist n > 0 such that
p+i
hi + u2\\PPXI + 2||mn2||4i < r, \\D
I|2(7+1)
U,
2
)
P+1 2
(1.27)
v+1
2 \ ~2~ ||2(7+1)
'Ui
2
,i=1
)
(1.28)
By definition of B wo get
2
E(t) ^
1
2(Y +1) i=1 1
EID
12(7+1)
1
2(y + 1) i=1 2
22 k„,.||2(7+1)__
2 p+1
p + 1 n
I i IIP+1 i oil |U1 + U2 Np+1 + 2|U1 U2
p±l 2
p+1 2
¿|D2(Y+1)1
p+1 2
(1.29)
i=1
1
2(Y +1) i=1
ENd'u
N2(y+1)
b (p+1) p + 1
eNd
N2(Y+1)
p+1 2
'Uil|2W ' ' I = /(a),
where a2 = ^2=1 ||DKui||2(7+1)- We can verify that the function / is increasing for 0 < a <
a1; decreasing for a > a1; / (a) ^ —œ as a ^ +œ, and
/ (a1) =
1
B (p+1)
-;-T Oi 1 — - LX-I
2(7 + 1) 1 p+1 1
ap+1 = d,
(1.30)
where a1 given in (1.21). Therefore, since E(0) < there exists a2 > a1 such that /(a2) = E(0). 2 1
Now we set a^ = ^2=1 ||DKui0H^1^, then by (1.29), we have /(a0) ^ E(0), which implies that a0 ^ a2. № establish (1.24), we suppose by contradiction that ||DKui(t0)|^(y+1) < a2 for some t0 > 0 to choose t0 such that ||DKui(t0)|2(7+1) > a2. Again using of (1.29) leads to
E(t0) > /( E |DKUi(t0)|2(7+1^ > /(a2) = E(0). ^ i=1 '
This is impossible since E(t) < E(0) (Vt G [0,T)).
E
1
EN"
N2(7+1)
2(y + 1) fe
Consequently, (1.24) gives
u,
2
< E(0) +
1
p + 1
I p+1 I |U1 + U2 Np+1 + 2|U1U2
p+1 2
p+1 2
p +1
I i NP+1 i nN |U1 + U2 |p+1 + 2|U1U2
p+1 2
p+1 2
2
> 2ÏÏTT) £ li™ilf+1) -^(0) > (Vi > 0). >
i=1
U
2
2
2
2
1
< Proof of Theorem 3.1. We set
H (t) = d — E (t).
By using (1.16), (1.31) we get
H/(t)^ (Mt)|k + |u2(t)|*) |u1(t)|2 dx
+ J (Mt)|e + Mt)|«) |u'2(t)|2 dx ^ 0 (Vt ^ 0). n
Therefore,
_ 2
0 < H(0) < H(t) =d- E \K
i=1
m
■ZID
KUi\\Ti+1) + 1
2(y +1) i=1" "*112 p +1 From (1.24), we obtain that for all t ^ 0 the estimates hold
M|u1 + u2 |p+1 +2||u1u2|
(p+l) 2
(p+l) 2
^-^TTyEII^^K
2(7+1) + 1
P + 1
M|u1 + u2|p+1 +2Mu1u2 1
(p+l) 2
(P+l) 2
<d
—-- Ofi -|--
2(7+1) 1 p+l
||u1 + u2 MPP+1 +2|u1u2 I
(p+l) 2
(p+l) 2
< --
1 2 1
a\ +
P + 1 1 P + 1
c0
||u1 + u2 MP+1 +2|u1u2|
(p+l) 2
(p+l) 2
<
p +1
M|u1 + u2|p+1 +2Mu1u2 1
(p+l) 2
(P+l) 2
Hence by (A2), we have
0 < H(0) < H(t) <
c1
E
ui
MP+1
p + 1 ^ II 'llp+r
i=1
Then we introduce
L(t) = H(t)+ ej ¿u,|ui|m-2uidx,
(1.31)
(1.32)
(1.33)
for e small to be chosen later and
'p — (k + 1) p — (l + 1) p — (g + 1) p — (0 +1) (p — (m — 1))
0 < a ^ min
p + 1 ' p + 1 ' m(p + 1)
p + 1 p + 1 We will show that L(t) satisfies
L/(t) ^ ^L1+v(t), for all t ^ 0, v > 0, f > 0,
(1.34)
(1.35)
1
2
1
1
defined in [0, to). By taking a derivative of (1.33) and using (1.1), we obtain
2
-H2(7+i)
llm 1 / y ||2
L/(t) = (1 - a)H(t)H/(t) + e ^ ||ui|m + ^ ID
i=i i=i
—e J ui ^|ui(t)|k + |u2(t)|'j ui dx — ej u2^|u2(t)|0 + |ui(t)|^ u'2dx n n
+e J (uifi(ui, u2) + u2f2(ui,u2)) dx.
Then
22
L/(t) = (1 — a)H(t)H/(t) + e £ ||u<||m + e E ID^I^
i=i i=i
—e ui (|ui (t)|k + |u2 (t)|^ ui dx — e u2 (ju2 (t)|0 + |ui(t)|^ u'2dx
11 P+i 11
+e ( 11ui + u2||p+i +2|uiu2
(p+i) 2
(p+1) 2
By exploiting (1.16) and (1.21), equation (1.36) takes the form
2
L'(t) > (1 - o)H-'(t)H'(t) + £m + 2(7 + 1)(m"1) £
m 2
V ||DKu,; i
—e / ui (|ui(t)|k + |u2(t)|^ uidx — e / u2 (|u2(t)|e + |ui(t)|^ u'2dx
m m
+e2(7 + 1)H(t) — e2(7 + 1)d + e2 ^ ||DKu|2(Y+i)
2(7 +1K A, , „ NP+i
+e(l--1 ) (J|«i + «2||p+1 + 2||mu2
(p+i) 2
(p+1) 2
We will estimate, for some constance Ai, A2 > 0, two terms as
|ui(t)|k + Mt)|^ |uiui|dx
Oi/ (Mi^ + Mi)!') |m|2dx + ^-J (Mi^ + Mi)!') K|2cte, nn
and
J (Mt)|0 + |ui(t)|^ |u2u2|dx n
<A2 J (|u2(i)|0 + |ui(i)|*) I U2\2dx + ^-J (|u2(i)|e + |ui(i)|*) \u'2\2 dx.
(1.36)
(1.37)
n
2
Then,
2
L'm mi - *)H-°w'(t)+i"^2'7;,1"""1' £ IK
i=1
+e2 ^ ||DKu,M2(Y+1) + 2(7 + 1)eH(t)
i=1
-eAi J (Mi^ + Mi)!') \Ul\2dx-e±-j (imWI^ + I^Wl') |ui|2dx n n
-eA2 J (|u2(i)|0 + Mir) \U2\2dx-e^-J (|n2(i)|0 + |m(i)|^) \u'2\2 dx.
and
nn Consequently, by using Young's inequality for some > 0, we have
J (Mt)|fc + |u2(t)|') |u1|2dx = ||u1 Mk+2 + J |u211M2dx nn
C11«. C+гг2г(,+2,/, + II"1 C
I (|«2(i)|' + M«|e) K|2dx = ||«2Mj+2 + / |«1|e|«2|2dx
nn
Then,
2
L'm mi - *)H-°w'(t)+/"+2'7;,1"""1' E11«:
2
+e2 ^ |DKui|2(Y+1) + 2(7 + 1)eH(t)
( ||u1 + u2 ||p+1 +2Mu1u2M
p±1 )
(M^ + M*)!') \u'2\2dx-e^-j (|m(i)|fc + \u2(t)\l) lu'^dx n n
(INK + ^-r^'lhllK + ^t"+2,/(2> IMS (11». K+jhs(,+2)" II"211«+rh2t-(,+2)m) 11«. K
Choosing A1; A2 such that
-J- = miH-a(t), -J- = m2H-a(t), mi,m2 > 0. (1.39)
4A1 4A2
2
Using (1.39) and the fact that
H'(t) = y (|U1 (í)|k + Mí)!') |ui(t)|2dx + J (|U2(í)|0 + |ui(i)|*) |u2(t)|2dx (Ví ^ 0),
n n
to obtain for M = m1 + m2 and assumption (A2),
Lj(m - IJ i 2
m
m > «1 - „) - Me)H-W'iO++ ')(■"-') V IK
m m
1
22 +e2 ^ ||DKUi|2(7+1) + 2(Y + (í) + ec^ HuiMP+1
i=1 i=1
-¿«-«I (IKC + ^r2,/'ih IIS+imi£ (ll«'K+rh^'lfoC+
Since (1.6) holds, we obtain by using condition (1.34)
ll¿+2 < Co I Hill
h ° (í)M«1 Mi+2 < C3 (||m n^;^ + ||U2|(¿+;- iN Mi+2J
Ha (Í)|U2 ||j+2 < C4 (|u2|-p(++)1)+(j+2) + ||U1 ||^(++)1) |U2 nj+2) , where i = k, 1, £ and j = 1. Then
2
m 5, ((1 _ _ Me)H-°WHt)+em+2{-i+^m-l) IK
i=1
22 +e2 ^ MDKUiM2(Y+1) + 2(7 + 1)eH(í) + ec^ ||ui|P+1 i=1 i=1
^¿"(ihllSír^ + ll»'»^"!!«»»«)
(lHK>1)+li,+2) + Ih IK," 11«. IIS)
(e+2)/(^ OKI"*"+ imO«»G
c 1 ; 1 ¿(1+2)n^ (\\vJa{p+1)+{l+2) ' IL..IIff(P+1)lL.-lli+2
,m2l+2t-{l+m2)^ (iMi;(;tr+2)+iwis^IWK) •
4m21 + 2 1 '+2
12
—e
(1.40)
By using (1.34) and the algebraic inequality
zv < (z + 1) < M + ij(z + a) (Vz ^ 0, 0 < v < 1, a ^ 0), (1.41)
we have, for all t ^ 0,
llu,||(p(++r'+2 < b (Mu,M(P+1) + H(0)) < b (Mu,M(P+1) + H(t)) , (1.42)
where b = 1 + 1/H(0) j = k, 0, l, g and i = 1, 2, so that we obtain
2
m > «1 _,) _ Jftun«)+e"'+2(7t„1>(""1> E11«:
m
i=1
2
+e2 ^ ||DKu,|2(Y+1) + 2(y + 1)eH(t) + ec^ |u-Mp+1
i=1 i=1
_ eJ_ C4 ffefiLJi^ + m^ + ii.jr^ii.ji^
4m1
c4 (b (Mu2M(P+1) + H(t)) + ||u1 MS(++)1) INK)
-¿¿r^'"« (* (||«i|iK!+ IMI^IMIK)
"^^TT1 MH^^+D+^V . irmoH-i) ll-lle+2
e
4m2
12
Also, since (X + Y)s ^ C(Xs + Ys), X, Y ^ 0, making use of (1.34) we conclude
(p+D-O+2) /
^ini^Tij-iKii^iKKij (L43)
(p+l)-0+2) / <7(p+l) + (j+2) CT(p+l) + (j+2)\ P+l 2
v y ,=1
Wliere C' = a{pllH(l+2) » C" = ' fOT J =
Similarly,
2
r(P+1) IL, Mj+2 / „ V^ |L, M(P+1)
(p(++)1) Mu2||j+2 < c6£ ||u,M(P+1), (1.44)
for j = 0, g, I.
Taking into account (1.43), (1.44), we deduce
L'(t) > ((1 - a) - Me)H-'(t)H'(t) + e™ +~ 1) £ ¡|n>
Then
2
m
+e2 ^ |DKui|2(7+i) + 2(7 + 1)eH(t) + ec^ ||ui|P+i
i=i
2
^¿«.ki^Ci+^+'-EWiSl!
2
i
(»(ii- +H(i))+«t ii-n:;::::,
(4 (INK!+*<«)+miss
--"¿^(KlI'-'llKi+H+^EII^IlK^
i=i
-B4r,jhr<'+1,mc3 (KlMCo+H +<»ElkCi
4m2 l + 2 _i
(1.45)
2
L'(t) > ((1 - a) - Me)H-°{t)H'it) + e™ ++ 1)(™ ~ £ |KC
i=i
+ £2E H^K^ll2(7+1) + £ (2(7 +1) + ^ ■C7 + ^ ■C8) ff (i) i=i
i=i
For large values of mi and m2 we can find positive constants ^^d B such that
L/(t) ^ ((1 — a) — Me)H(t)H/(t)
(1.46)
:,„ + 2(7;i,(m-1)E||<||:;+^(i)+gBE|K|C, (1.47)
We pick e small enough so that ((1 — a) — Me) ^ ^d L (0) > 0. Consequently, there exists r > 0 such that
L/(t) > er H(t) + ^ ||ui|m + E ||ui|(P+i) . (1.48)
m
2
2
Thus, we have L(t) ^ L(0) > 0, for all t ^ 0. On the other hand, we have
L^(t)= iH^ty+e I J2ui\ui\m~2ui(x^)dx
i
1 -CT
< do H (t) +
E«iK |m-1 dx
1
1 — CT
By Hdlder's and Young's inegualities, taking (1.6) into accaunt, we estimate
- - M .. - ..... - - 1 1
uilu^r^dx ^ Ilm u-
m II Him
m — mo" I (1 —ct) (1 —m<r)
lp+1
+ llu
/ llm
m
(1.49)
i = 1,2,
and also using (1.34), we have
i
1 — CT
E«iKr-1 " < / uiiuii
/ im-1
i=1
1
l-CT
+ J U2|u2|
n
/ im-1
1
l-CT
<
C EIKIIpVTCT)+EIK
By using again (1.34) and (1.41) we get
I (1 — ma) l(p+1)
Therefore,
H(t) + £ MuiM(P+1) + E IK
1 (p+1) 1 ^^ N^llm
= 1 i=1
(Vt ^ 0).
With (1.52) and (1.48), we arrive at
i
L'(t) ^ a0L^(t) (Vi^O). Finally, a simple integration of (1.53) gives the desired result. >
(1.50)
< 6 (lMl(p+1) + H(t)) (i = 1,2, Vt ^ 0). (1.51)
(1.52)
(1.53)
2
i
u
2
2
2
iii
m
m
iii
u
2
2
m
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 /1; /2 drive the solution of our system to blow up tp 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
ui/ — 0 (||Vui 112) Aui + ^ (||Vui ||2^ gi (t — s)Aui(s) ds = 0,
0t (1-54)
u/2/ — 0 (||Vu2||2) Au2 + ^ (|Vu2|2) / g2(t — s)Au2(s) ds = 0,
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.
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Received March 19, 2014-
Khaled Zennir Department of Mathematics, College of Sciences and Arts, Al-Ras, Al-Qassim University, Kingdom of Saudi Arabia;
Laboratory of Lamahis, Department of Mathematics University 20 Août 1955, Skikda 21000, Algeria Email:khaledzermir2<3yahoo. com
Salah Zitouni Department of Mathematics, University Badji Mokhtar Annaba 23000, Algeria Email:zitsal@yahoo. fr
НЕСУЩЕСТВОВАНИЕ РЕШЕНИЯ ЗАТУХАЮЩЕЙ СИСТЕМЫ НЕЛИНЕЙНЫХ ВОЛНОВЫХ УРАВНЕНИЙ ТИПА КИРХГОФА
Зеннир К., Зитуни С.
Изучается влияние сильного источника на существование решений в пространстве с высоким порядком суммируемости в затухающей системе нелинейных волновых уравнений типа Кирхгофа.
Ключевые слова: взрыв, уравнение типа Кирхгофа, вырождающиеся затухающие системы, сильно нелинейный источник, положительная начальная энергия.
