J Dyn Control Syst DOI 10.1007/s10883-015-9280-9
A Blow-up Result in a Viscoelastic System Mohammad Kafini1
Received: 9 November 2014 / Revised: 14 April 2015 © Springer Science+Business Media New York 2015
Abstract In this paper, we consider a system of nonlinear viscoelastic wave equations. We shall show that the global nonexistence results for this system found in the literature can be extended to a bigger region. Keywords Viscoelastic · Global nonexistence · Wave system · Blow up Mathematics Subject Classification (2010) 35B37 · 35L55 · 74D05 · 93D15 · 93D20
1 Introduction Viscoelastic systems with single equation were considered by many authors. For example, Messaoudi in [1] considered the following initial-boundary value problem ⎧ t ⎨ utt − u + 0 g(t − τ )u(τ )dτ + ut |ut |m−2 = u|u|p−2 , in × (0, ∞) (1) u(x, t) = 0, x ∈ ∂ , t ≥ 0 ⎩ u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ . where is a bounded domain of Rn (n ≥ 1) with a smooth boundary ∂, p > 2, m ≥ 1, and g : R+ −→ R+ is a positive nonincreasing function. He showed, under suitable conditions on g, that solutions with initial negative energy blow up in finite time if p > m and continue to exist if m ≥ p. This result has been later pushed, by the same author [2], to
Mohammad Kafini
[email protected] 1
Department of Mathematics and Statistics, KFUPM, Dhahran 31261, Saudi Arabia
Mohammad Kafini
certain solutions with positive initial energy. A similar result has also been obtained by Wu [3] using a different method. In the absence of the viscoelastic term (g = 0), problem (1) has been extensively studied and many results concerning global existence and nonexistence have been proved. For instance, for the equation utt − u + aut |ut |m = b|u|γ u,| in × (0, ∞)
(2)
m, γ ≥ 0, it is well known that, for a = 0, the source term bu|u|γ , (γ > 0) causes finite time blow up of solutions with negative initial energy (see [4]). The interaction between the damping and the source terms was first considered by Levine [5, 6] in the linear damping case (m = 0). He showed that solutions with negative initial energy blow up in finite time. Georgiev and Todorova [7] extended Levine’s result to the nonlinear damping case (m > 0). In their work, the authors introduced a different method and showed that solutions with negative energy continue to exist globally “in time” if m ≥ γ and blow up in finite time if γ > m and the initial energy is sufficiently negative. This last blow-up result has been extended to solutions with negative initial energy by Messaoudi [8] and others. For results of same nature, we refer the reader to Levine and Serrin [9], Vitillaro [10], and Messaoudi and Said-Houari [11]. For problem (2) in Rn , we mention, among others, the work of Levine Serrin and Park [12], Todorova [13, 14], Messaoudi [15], and Zhou [16]. Recently, Autuori et al. [17] investigated a global nonexistence for nonlinear Kirchhoff system. They established their result using the new approach of the classical potential well and the concavity method when the initial energy is controlled above by a critical value. Motivated by the above works, Wu and Lin [18] showed that the global nonexistence results for t g(t − s)u(x, s)ds + Q(x, t, ut ) = f (x, u), in × (0, ∞), utt − u + 0
can be extended to a bigger region. In this work, we are concerned with the following system of nonlinear viscoelastic wave equations t utt − u + 0 g(t − s)u(x, s)ds + ut |ut |m−2 = f1 (u, v), in × (0, ∞) t vtt − v + 0 h(t − s)v(x, s)ds + vt |vt |r−2 = f2 (u, v), in × (0, ∞) (3) u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ , v(x, 0) = v0 (x), vt (x, 0) = v1 (x), x ∈ , where g, h, u0 , u1, v0 , v1 are functions to be specified later and p
p
p
p
f1 (u, v) = a |u + v|p−2 (u + v) + b |u| 2 −2 u |v| 2 , f2 (u, v) = a |u + v|p−2 (u + v) + b |v| 2 −2 v |u| 2 ,
(4)
u = u(x, t), v = v(x, t), t ∈ R+ , x ∈ a bounded domain of R N (N ≥ 1) with a smooth boundary ∂. The constants p, m and r satisfying 2 (N − 1) if N ≥ 3 2 < p if N = 1, 2 and 2 < p ≤ N −2 and 2 < r, m if N = 1, 2
and 2 < r, m ≤
N +2 N −2
if N ≥ 3.
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A Blow-up Result in a Viscoelastic System
This type of problems arises in viscoelasticity and in systems governing the longitudinal motion of a viscoelastic configuration obeying a nonlinear Boltzmann’s model. Different authors worked on this type of systems. Global existence and nonexistence were established (see, for instance, [19], [20], and [21]). Our aim here is to extend the result of [18], established for the wave equation, to our system. We shall show that the global nonexistence results for equations in Eq. 3 found in the literature can be extended. For instance, the blow up results found for E(0) < E1 and a region = {(λ, E) | λ > λ1 , %E < E1 } . 1 , In our work, we will see that this region can be extended to by taking E(0) < E 1 > E1 . Hence, the new extended region takes the form for E
1 . = (λ, E) | λ > λ1 , E < E To achieve this goal, some conditions have to be imposed on the initial data, relaxation functions g and h, and the sources f1 and f2 . The contents of this paper is organized as follows. In Section 2, we state the local existence result in addition to conditions imposed to the relaxation and the source functions. In Section 3, we state and prove our main result.
2 Preliminaries In this section, we prepare some material needed in the proof of our main result. For this reason, we assume that (G1) g, h : R+ −→ R+ are differentiable functions such that
∞
1− 0
1−
∞
g(s)ds = l > 0,
g (t) ≤ 0,
g(0) > 0, t ≥ 0.
h(s)ds = k > 0,
h (t) ≤ 0,
h(0) > 0, t ≥ 0.
0
Remark (G1) is necessary to guarantee the hyperbolicity of the system (3). Lemma 2.1 [19]. For F (u, v) =
p 1 a |u + v|p + 2b |uv| 2 , p
we have uf1 (u, v) + vf2 (u, v) = pF (u, v) and, for some, d0 , d1 > 0, d1 p d0 p |u| + |v|p ≤ F (u, v) ≤ |u| + |v|p . p p
(6)
Corollary 2.1 As a result of Lemma 2.1 [19], there exists ε0 > 0 such that for all ε ∈ (0, ε0 ), there exists d2 > 0 depending on ε such that uf1 (x, u) + vf2 (x, v) − (p − ε) F (u, v) ≥ d2 |u|p + |v|p ,
∀u, v ∈ H01 ().
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Mohammad Kafini
Lemma 2.2 [19]. There exists d3 > 0 such that, ∀x ∈ ,
|fi (u, v)| ≤ d3 |u|p−1 + |v|p−1 for
i = 1, 2,
We introduce the “modified” energy functional E(t) : =
t t 1 1 1 1 g(s)ds ∇u 22 + h(s)ds ∇v 22 ut 22 + vt 22 + 1− 1− 2 2 2 2 0 0 1 1 + (g ◦ ∇u) + (h ◦ ∇v) − F (u, v)dx, t ≥ 0, (8) 2 2
where
t
(g ◦ u)(t) =
0 t
(h ◦ v)(t) = 0
g(t − τ ) u(t) − u(τ ) 22 dτ, h(t − τ ) v(t) − v(τ ) 22 dτ.
Also, we set −1 λ1 = c∗ d1 B p p−2 , 1 w1 = λ21 , 2 −2 2 1 ∗ 1 c d1 B p p−2 = 1 − − w1 , E1 = 2 p p where B = given by
√ B1 , min{l,k}
(9)
B1 is the best constant of the Sobolev embedding H01 () → Lp () p B1−1 = inf ∇u 22 : u ∈ H01 (), u p = 1 ,
and c∗ is a positive constant, depending on p, which controls the inequality p for a, b ≥ 0. a + bp ≤ c∗ (a + b)p , On the other hand, from Eq. 8 and recalling Eq. 14, we have 1 1 F (u, v)dx = ut 22 + vt 22 2 2 t t 1 1 1− 1− + g(s)ds ∇u 22 + h(s)ds ∇v 22 2 2 0 0 1 1 + (g ◦ ∇u) + (h ◦ ∇v) − E(t) 2 2 k l 2 (10) ≥ ∇u 2 + ∇v 22 − E(t) ≥ −E(0), 2 2 which means that F (u, v)dx is bounded below by −E(0) for any solution (u, v) and for all t ≥ 0. Therefore, we may assume that w2 = inf F (u, v)dx > −∞. (11) t≥0
A Blow-up Result in a Viscoelastic System
The local existence result for problem (3) is stated in the following proposition. The proof of this result was given in [20], in which the authors adopted the technique of [21] which consists of constructing approximations by the Faedo–Galerkin procedure without imposing the usual smallness conditions on the initial data in order to handle the source terms. Due to the strong nonlinearities on f1 and f2 , the techniques used allowed them to prove the local existence result only for N ≤ 3. Proposition 2.1 Let N ≤ 3. Assume that (G1) holds. If ((u0 , v0 ), (u1 , v1 )) ∈ H01 () × L2 (), then there exists a unique weak local solution of Eq. 3 such that, for some T > 0, (u, v) ∈ C [0, T ), H01 () , and
ut ∈ C [0, T ) , L2 () ∩ Lm+1 ( × [0, T )) , vt ∈ C [0, T ) , L2 () ∩ Lr+1 ( × [0, T )) .
3 Blow Up In this section, we state and prove our main result. Theorem 3.1 Assume that p > max {2, m, r} and (G1) holds. Assume further that
∞
max
g(s)ds, 0
∞
h(s)ds ≤
0
d2 . 2d1 + d2
(12)
Then, for any initial data ((u0 , v0 ), (u1 , v1 )) ∈ H01 () × L2 () satisfying 1 , E(0) < E
(13)
where
1 = p − 1 w2 , E 2 the corresponding solution of Eqs. 3–5 blows up in finite time. In order to establish our proof, we need the following lemmas. Lemma 3.2 If (u,v) is a solution of Eq. 3, then E(t) is a nonincreasing function on [0, T ] and 1 1 1 1 g ◦ ∇u + h ◦ ∇v − g(s) ∇u 22 − h(s) ∇v 22 E (t) = 2 2 2 2 1 1 m r − ut m − vt r ≤ 0. (14) 2 2 Proof By multiplying the equations in Eq. 3 by ut and vt , respectively, integrating over , using integration by parts, and repeating the same computations as in [19], we obtain the result. Hence, E(t) ≤ E(0).
Mohammad Kafini
Lemma 3.3 Let (u,v) be any solution of Eq. 3, then 1
2 2 2 l ∇u 2 + k ∇v 2 + (g ◦ ∇u) + (h ◦ ∇v) , E(t) ∈ where
1 . = (λ, E) | λ > λ1 , E < E
If, furthermore, Eq. 13 holds, then w2 > 0. Proof Using Eq. 11, we deduce from Eq. 13 that p F (u, v)dx ≥ −E(0) > − w2 = inf − 1 w2 , 2 that is w2 > 0. To prove the first part of Lemma 3.3, using Eq. 6, we estimate p d1 |u| + |v|p dx p d1 p p u p + v p ≤ p d1 p/2 p p ∇u 2 + ∇v 2 B ≤ p 1 p d1 ∗ p/2 2 ≤ ∇u 22 + ∇v 22 c B1 p p d1 ∗ p/2 2 c B ≤ l ∇u 22 + k ∇v 22 p p d1 ∗ p/2 2 l ∇u 22 + k ∇v 22 + (g ◦ ∇u) + (h ◦ ∇v) , c B ≤ p and assume, for contradiction, that there exists t1 > 0 such that 1 2 l ∇u(t1 ) 22 + k ∇v(t1 ) 22 + (g ◦ ∇u)(t1 ) + (h ◦ ∇v)(t1 ) ≤ λ1 .
F (u, v)dx ≤
(15)
(16)
So, by Eqs. 8, 13, and 15, we have c∗ d B p/2 p 2 1 l ∇u(t1 ) 22 + k ∇v(t1 ) 22 + (g ◦ ∇u)(t1 ) + (h ◦ ∇v)(t1 ) 2 p p p 1 > E(0) −1 − 1 w2 = E ≥ F (u, v)dx ≥ 2 2 1 l ∇u(t1 ) 22 + k ∇v(t1 ) 22 + (g ◦ ∇u)(t1 ) + (g ◦ ∇v)(t1 ) ≥ 2 − F (u, v)(t1 )dx p
−1
1 l ∇u(t1 ) 22 + k ∇v(t1 ) 22 + (g ◦ ∇u)(t1 ) + (g ◦ ∇v)(t1 ) ≥ 2 p c∗ d1 B p/2 2 l ∇u(t1 ) 22 + k ∇v(t1 ) 22 + (g ◦ ∇u)(t1 ) + (g ◦ ∇v)(t1 ) , − p
A Blow-up Result in a Viscoelastic System
we conclude that −1 1 2 p−2 = λ1 , l ∇u (t1 ) 22 + k ∇v (t1 ) 22 + (g ◦ ∇u) (t1 ) + (g ◦ ∇v) (t1 ) > c∗ d1 B p/2 (17) which contradicts the assumption (16). So, we conclude that p 2 > λ1 , l ∇u(t1 ) 22 + k ∇v(t1 ) 22 + (g ◦ ∇u)(t1 ) + (g ◦ ∇v)(t1 ) which means that 1
2 2 2 . l ∇u 2 + k ∇v 2 + (g ◦ ∇u) + (g ◦ ∇v) , E(t) ∈ Now, we are in position to start the proof of the main theorem. Proof of Theorem 3.1 We set H (t) = E2 − E(t), where
t ≥ 0,
1 . E2 > 0 such that E2 ∈ E(0), E
We easily can see that H (t) ≥ 0. Thus, H (t) ≥ H (0) = E2 − E(0) > 0,
t ≥ 0.
(18)
Moreover, estimates Eqs. 6, 8, 13, and 11 yield 1 + F (u, v)dx H (t) = E2 − E(t) < E p p F (u, v)dx ≤ F (u, v)dx − 1 w2 + = 2 2 d1 p p u p + v p . ≤ 2 Let
A(t) =
(uut + vvt ) dx.
Using Eq. 3, we find A (t) = ut 22 + vt 22 − ∇u 22 − ∇v 22 t + g(t − s)∇u(x, t).∇u(x, s)dsdx 0 t h(t − s)∇v(x, t).∇v(x, s)dsdx + 0 − u|ut |m−2 ut + v|vt |m−2 vt dx + (uf1 (u, v) + vf2 (u, v)) dx.
(19)
Mohammad Kafini
Using Young’s inequality, we estimate
t
g(t − s)∇u(t).∇u(s)dsdx
0
t
g(t − s)∇u(t). (∇u(s) − ∇u(t)) dsdx + 0 1 1 t ≥ − (g ◦ ∇u)(t) + g(s)ds ∇u(t) 22 , 2 2 0
=
t 0
g(t − s)ds ∇u(t) 22
also, 0
t
1 1 h(t − s)∇v(t).∇v(s)dsdx ≥ − (h ◦ ∇v)(t) + 2 2
t 0
h(s)ds ∇v(t) 22 .
Thus,
1 1 A (t) ≥ ut 22 + vt 22 − ∇u 22 − ∇v 22 − (g ◦ ∇u)(t) − (h ◦ ∇v)(t) 2 2 1 t 1 t 2 2 g(s)ds ∇u(t) 2 + h(s)ds ∇v(t) 2 + 2 0 2 0 m−2 m−2 − ut + v|vt | vt dx + u|ut | (uf1 (u, v) + vf2 (u, v)) dx. (20)
From Eq. 7, we fix ε0 small—if needed—so that ε0 w2 ≤ (p − 2) w2 − 2E2 ,
(21)
1 . Then, we choose ε ∈ (0, ε0 ) to get, from which is possible because w2 > 0 and E2 < E Eqs. 8 and 20,
A (t) ≥
p−ε+2 ut 22 + vt 22 − (p − ε) E(t) 2 t t 1 1 1− 1− + (p − ε − 2) g(s)ds ∇u 22 + h(s)ds ∇v 22 2 2 0 0 1 1 u|ut |m−2 ut + v|vt |m−2 vt dx + (p − ε − 2) (g ◦ ∇u) + (g ◦ ∇v) − 2 2 + F (u, v)dx (uf1 (u, v) + vf2 (u, v)) dx − (p − ε) 1 1 t 1 t 1 g(s)ds ∇u 22 − h(s)ds ∇v 22 . + (g ◦ ∇u) + (h ◦ ∇v) − 2 2 2 0 2 0
A Blow-up Result in a Viscoelastic System
Using Eqs. 7 and 8 again, we obtain
A (t) ≥
p−ε+2 ut 22 + vt 22 2 t t 1 1 g(s)ds ∇u 22 + h(s)ds ∇v 22 1− 1− + (p − ε − 2) 2 2 0 0 1 1 p p + (g ◦ ∇u) + (g ◦ ∇v) − E(t) − 2E(t) + d2 u(t) p + v(t) p 2 2 − u|ut |m−2 ut + v|vt |m−2 vt dx 1 1 t 1 1 t + (g ◦ ∇u) + (h ◦ ∇v) − g(s)ds ∇u 22 − h(s)ds ∇v 22 . 2 2 2 0 2 0
Using Eqs. 8 and 11, we obtain p p A (t) ≥ 2 ut 22 + vt 22 + d2 u(t) p + v(t) p + (p − ε − 2) w2 − 2E2 + 2H (t) 1 1 t 1 t 1 2 g(s)ds ∇u 2 − h(s)ds ∇v 22 + (g ◦ ∇u) + (h ◦ ∇v) − 2 2 2 0 2 0 u|ut |m−2 ut + v|vt |m−2 vt dx. −
Since ε < ε0 , we have from Eq. 21 that (p − ε − 2) w2 − 2E2 > 0 and p p A (t) ≥ 2 ut 22 + vt 22 + d2 u(t) p + v(t) p + 2H (t) 1 1 t 1 1 t + (g ◦ ∇u) + (h ◦ ∇v) − g(s)ds ∇u 22 − h(s)ds ∇v 22 2 2 2 0 2 0 − u|ut |m−2 ut + v|vt |r−2 vt dx. (22)
To estimate the last term of Eq. 22, we use Young’s inequality
XY ≤
δ r r δ −q q X + Y , r q
X, Y ≥ 0,
∀δ > 0,
1 1 + = 1. r q
By taking r = m and q = m/(m − 1), we have |u||ut |m−1 dx ≤
δm m − 1 −m/(m−1) ut m u m δ m+ m. m m
Similarly, |v||vt |r−1 dx ≤
δr r − 1 −r/(r−1) vt rr . v rr + δ r r
Mohammad Kafini
Thus, Eq. 22 becomes p p A (t) ≥ 2 ut 22 + vt 22 + d2 u(t) p + v(t) p + 2H (t) 1 t 1 1 t 1 g(s)ds ∇u 22 − h(s)ds ∇v 22 + (g ◦ ∇u) + (h ◦ ∇v) − 2 2 2 0 2 0 δm m − 1 −m/(m−1) δr r − 1 −r/(r−1) − u m ut m vt rr . δ v rr − δ m− m− m m r r Then, we define L(t) = H 1−σ (t) + δ1 A(t),
t ≥ 0.
where δ1 > 0 to be defined later and 0 < σ < min
p−r p−2 p−m , , . p(m − 1) p(r − 1) 2p
(23)
A direct differentiation of L(t) using A (t), give 4δ1 + k ut 22 + vt 22 L (t) ≥ (1 − σ )H (t)H (t) + (2δ1 + k) H (t) + 2 t δ1 + k k δ1 + k g(s)ds ∇u 22 + − ((g ◦ ∇u) + (g ◦ ∇v)) + 2 2 2 0 t δ1 + k k 2 − + g(s)ds ∇u 2 − kE2 − k F (u, v)dx 2 2 0 p p +δ1 d2 u(t) p + v(t) p m −δ m − 1 −m/(m−1) δr r − 1 −r/(r−1) m m r r ut m − v r − vt r , u m − δ δ +δ1 m m r r (24)
−σ
for some positive constant k. From Eq. 6 and 11, we note that 1 = E2 < E
p p d 1 p p F (u, v)dx ≤ u(t) p + v(t) p , − 1 w2 ≤ −1 −1 2 2 2 p
p
thus, for k =
δ1 d2 2d1 ,
we estimate p p δ1 d2 u(t) p + v(t) p − kE2 − k kd1 p p u(t) p + v(t) p ≥ δ1 d2 − 2 3δ1 d2 p p u(t) p + v(t) p ≥ 0. = 4
F (u, v)dx
(25)
A Blow-up Result in a Viscoelastic System
Combining Eqs. 24 and 25, give L (t)
8d1 δ1 + δ1 d2 4d1 δ1 + δ1 d2 ≥ (1 − σ )H −σ (t)H (t) + H (t) + ut 22 + vt 22 2d1 4d1 t δ1 d2 2d1 δ1 + δ1 d2 2d1 δ1 + δ1 d2 − g(s)ds ∇u 22 + ((g ◦ ∇u) + (h ◦ ∇v)) + 4d1 4d1 4d1 0 t δ1 d2 2d1 δ1 + δ1 d2 δ1 d2 p p u(t) p + v(t) p + − h(s)ds ∇v 22 + 4d1 4d1 4 0 m m − 1 −m/(m−1) δr r − 1 −r/(r−1) −δ m m r u m − δ δ ut m − v r − vt rr . +δ1 m m r r
Exploiting assumption (12), leads to L (t)
4d1 δ1 + δ1 d2 8d1 δ1 + δ1 d2 H (t) + ut 22 + vt 22 ≥ (1 − σ )H −σ (t)H (t) + 2d1 4d1 δ1 d 2 2d1 δ1 + δ1 d2 p p u(t) p + v(t) p + ((g ◦ ∇u) + (h ◦ ∇v)) + 4d1 4 m −δ m − 1 −m/(m−1) δr r − 1 −r/(r−1) m r r − u − − v u m δ v δ (26) +δ1 t m t r m r m m r r As in [8], estimation (26) remains valid even if δ is time dependent. Therefore, taking δ so that (27) min δ −m/(m−1) , δ −r/(r−1) = k1 H −σ (t), for large k1 to be specified later, we estimate k11−m σ (m−1) k11−r σ (r−1) δr δm r m u m v H H + ≤ (t) u + (t) v rr . m r m m r m r m r r Exploiting Eq. 19 and the inequalities u m m ≤ C u p and v r ≤ C v p , we obtain
σ (m−1) d1 p p u p + v p u m p 2 σ (m−1) d1 m+σp(m−1) pσ (m−1) u p ≤ C + v p u m p . 2
H σ (m−1) (t) u m m ≤
Using Young’s inequality for q=
m + σp(m − 1) , pσ (m − 1)
we estimate, for C > 0, pσ (m−1)
v p
q =
m + σp(m − 1) , m
m+σp(m−1) m+σp(m−1) . u m + u p p ≤ C v p
Mohammad Kafini
Thus,
m+σp(m−1) m+σp(m−1) m+σp(m−1) + v p + u p H σ (m−1) (t) u m . m ≤ C u p
Similarly,
r+σp(r−1) r+σp(r−1) r+σp(r−1) + u p + v p H σ (r−1) (t) v rr ≤ C v p .
Using Eq. 23 and Corollary 2.2 [8] again, we have m+σp(m−1)
r+σp(r−1)
+ v p u p p p ≤ C −H (t) − ut 22 − vt 22 − (g ◦ ∇u) − (h ◦ ∇v) + u p + v p . Therefore, δr δm u m v rr m+ m r ∗ p p ≤ C1 k11−m −H (t) − ut 22 − vt 22 − (g ◦ ∇u) − (g ◦ ∇v) + u p + v p .(28) for some positive constant C1 and m∗ = min {m, r} ≥ 2. Moreover, using Eqs. 14 and 27 and recalling that H (t) is stands for −E (t), give r − 1 −r/(r−1) m − 1 −m/(m−1) δ δ ut m vt rr ≥ −C2 k1 H −σ (t)H (t). m− m r Combining Eqs. 26–29, we obtain −
L (t)
4d1 + d2 1−m∗ + C1 k1 H (t) ≥ (1 − σ − δ1 C2 k1 )H (t)H (t) + δ1 2d1 ∗ 8d1 + d2 +δ1 + C1 k11−m ut 22 + vt 22 4d1 ∗ 2d1 + d2 +δ1 + C1 k11−m ((g ◦ ∇u) + (g ◦ ∇v)) 4d1 d2 p p 1−m∗ − C1 k1 u p + v p . +δ1 4 −σ
(29)
At this point, we choose k1 large enough so that we pick δ1 small enough so that
d2 4
(30)
∗
− C1 k11−m > 0. Once k1 is fixed,
1 − σ − δ1 C2 k1 > 0
and L(0) = H 1−σ (0) + δ1
(u0 u1 + v0 v1 ) dx > 0.
Therefore, Eq. 30 takes the form p p L (t) ≥ δ1 γ H (t) + ut 22 + vt 22 + (g ◦ ∇u) + (h ◦ ∇v) + u p + v p , where γ is the minimum of coefficients in Eq. 31. Consequently, we have L(t) ≥ L(0) > 0,
t ≥ 0.
(31)
A Blow-up Result in a Viscoelastic System
Next, we have, by Holder’s and Young’s inequalities 1/(1−σ )
(uut + vvt )dx
≤ 2
1/(1−σ )
≤ C
1/(1−σ ) 1/(1−σ ) uut dx + vvt dx
μ/(1−σ ) u p
μ/(1−σ ) + v p
θ/(1−σ )
+ ut 2
θ/(1−σ )
+ vt 2
,
(32)
for 1/μ + 1/θ = 1. To be able to use Corollary 2.6 [8], we take θ = 2(1 − σ ) which gives μ/(1 − σ ) = 2/(1 − 2σ ) ≤ p. Therefore, for s = 2/(1 − 2σ ), Eq. 30 becomes 1/(1−σ ) (uut + vvt )dx ≤ C u sp + v sp + ut 22 + vt 22 .
So, Corollary 2.2 [8], gives 1/(1−σ ) (uut + vvt )dx
p p ≤ C H (t) + ut 22 + vt 22 + (g ◦ ∇u) + (h ◦ ∇v) + u p + v p .
(33)
Therefore, we have 1/(1−σ ) L1/(1−σ ) (t) = H 1−σ (t) + δ1 A(t) 1/(1−σ ) ! 1/(1−σ ) H (t) + δ1 ≤ 2 (uut + vvt )dx
p p ≤ C H (t)+ ut 22 + vt 22 +(g ◦ ∇u)+(h ◦ ∇v)+ u p + v p ,
t ≥ 0. (34)
Combining Eqs. 31 and 34, we arrive at L (t) ≥ L1/(1−σ ) (t),
t ≥ 0.
(35)
where is a positive constant depending only on δ1 γ and C. A simple integration of Eq. 35 over (0, t) yields 1 . Lσ/(1−σ ) (t) ≥ −σ/(1−σ ) L (0) − σ t/(1 − σ ) Therefore, L(t) blows up in time T ≤ T∗ =
1−σ
σ L−σ/(1−σ ) (0)
.
1 . This shows the result for E(0) < E If, further, E(0) < E1 , then by Eq. 17, we have 1 1 l ∇u 22 + k ∇v 22 + (g ◦ ∇u) + (g ◦ ∇v) > λ21 = w1 . 2 2
(36)
Mohammad Kafini
Exploiting Eqs. 10 and 36, we have 1 l ∇u 22 + k ∇v 22 + (g ◦ ∇u) + (h ◦ ∇v) − E(0) F (u, v)dx ≥ 2 ≥ w 1 − E1 . This yields that w2 ≥ w1 − E1 and, using Eq. 9, 1 = p − 1 w2 = p 1 − 2 w2 > p 1 − 2 (w1 − E1 ) = E1 . E 2 2 p 2 p Hence, we extended the nonexistence region from = {(λ, E) | λ > λ1 , to
= (λ, E) | λ > λ1 ,
E < E1 } 1 . E
This completes the proof. Acknowledgments The author would like to express his sincere thanks to King Fahd University of Petroleum and Minerals for its support. This work has been funded by KFUPM under Project No. IN 111024.
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