Open Access

Global existence and blow-up of solutions for a nonlinear wave equation with memory

Journal of Inequalities and Applications20122012:33

https://doi.org/10.1186/1029-242X-2012-33

Received: 12 June 2011

Accepted: 15 February 2012

Published: 15 February 2012

Abstract

In this article, we consider the nonlinear viscoelastic equation

u t t - Δ u + 0 t g ( t - τ ) Δ u ( τ ) d τ - ω Δ u t + μ u t = u p - 2 u

with initial conditions and Dirichlet boundary conditions. We first prove a local existence theorem and show, for some appropriate assumption on g and the initial data, that this solution is global with energy which decays exponentially under the potential well. Secondly, not only finite time blow-up for solutions starting in the unstable set is proved, but also under some appropriate assumptions on g and the initial data, a blow-up result with positive initial energy is established. Finally, we also prove the boundedness of global solutions for strong (ω > 0) damping case.

2000 MSC: 35L05; 35L15; 35L70.

Keywords

global existence blow-up wave equation memory

1. Introduction

In this article we study the behavior of solutions for the following nonlinear viscoelastic equation
u t t - Δ u + 0 t g ( t - τ ) Δ u ( τ ) d τ - ω Δ u t + μ u t = u p - 2 u , x Ω , t > 0 , u ( x , t ) = 0 , x Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) x Ω ,
(1.1)
where Ω is a bounded domain in n with a smooth boundary ∂Ω, g is a positive function satisfying some conditions to be specified later, ω, μ satisfy
ω 0 , μ > - λ ω ,
(1.2)
λ being the first eigenvalue of the operator -Δ under homogeneous Dirichlet boundary conditions, and
2 < p 2 n n - 2 , for ω > 0 , 2 n - 2 n - 2 , for ω = 0 , if n 3 , 2 < p < if n = 1 , 2 .
(1.3)

This problem has its origin in the mathematical description of viscoelastic materials. It is well known that viscoelastic materials exhibit natural damping, which is due to the special property of these materials to retain a memory of their past history. A general theory concerning problem (1.1) in the case ω = 0 and μ = 0 is available in literature (see [14]). The asymptotic behavior of the solutions to (1.1) has been studied in [58], we also refer to [9, 10] for the asymptotic decay of the solutions to problems analogous to (1.1). Among other known results about problem (1.1) with ω = 0 and μ = 0, we recall that in [7, 8], it is proved that the exponential decay of g is a sufficient condition to the exponential decay of the solution u. In [5] it is also proved that, when ω = 0 and μ = 0, the exponential decay of g is necessary for the exponential decay of u. When ω + μ ≠ 0, Fabrizio and Polidoro [11] showed that the exponential decay of g is a necessary condition for the exponential decay of u. The case of only having 0 t g ( t - τ ) Δ u ( τ ) d τ may be very restrictive in many physical problems. Also, problem (1.1) is applied to the theory of the heat conduction with memory, see [1216]. Therefore, the dynamics of (1.1) are of great importance and interest as they have wide applications in natural sciences.

This type of problem have been considered by many authors and several results concerning existence, nonexistence, and asymptotic behavior have been established. Cavalcanti et al. [17] studied the following equation:
u t t - Δ u + 0 t g ( t - τ ) Δ u ( τ ) d τ + a ( x ) u t + u γ u = 0 , in Ω × ( 0 , )
for a : Ω → +, a function, which may be null on a part of the domain Ω. Under the conditions that a(x) ≥ a0 > 0 on Ω1 Ω, with Ω1 satisfying some geometry restrictions and
- ξ 1 g ( t ) g ( t ) - ξ 2 g ( t ) , t 0 ,
the authors established an exponential rate of decay. This latter result has been improved by Cavalcanti and Oquendo [18] and Berrimi and Messaoudi [19]. In their work, Cavalcanti and Oquendo [18] considered the situation where the internal dissipation acts on a part of Ω and the viscoelastic dissipation acts on the other part. They established both exponential and polynomial decay results under conditions on g and its derivatives up to the third order, whereas Berrimi and Messaoudi [19] allowed the internal dissipation to be nonlinear. They also showed that the dissipation induced by the integral term is strong enough to stabilize the system and established an exponential decay for the solution energy provided that g satisfies a relation of the form
g ( t ) - ξ g ( t ) , t 0 .

In [20], Berrimi and Messaoudi considered problem (1.1) for ω = μ = 0. They established a local existence result and showed, for certain initial data and suitable conditions on g, that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g.

For nonexistence, we should mention that Messaoudi [21] looked into the equation
u t t - Δ u + 0 t g ( t - τ ) Δ u ( τ ) d τ + u m - 2 u = u p - 2 u , in Ω × ( 0 , )
(1.4)

and proved, under appropriate relations between p, m and g, a blow-up result. This work generalizes earlier ones by Georgiev and Todorova [22] and Messaoudi [23], in which a similar result has been established for the wave equation (g ≡ 0). This result was later improved by Messaoudi [24], to certain solutions with positive initial energy. A similar result was also obtained by Wu [25] using a different method. For the problem (1.4) in n and with m = 2, Kafini and Messaoudi [26] showed, for suitable conditions on g and initial data, that solutions with negative energy blow up in finite time. More recently, Wang [27] has investigated a sufficient condition of the initial data with arbitrarily positive initial energy such that the corresponding solution of Equation (1.4) with m = 2 blows up in finite time. This result improved the blow-up results in [21, 24].

In this article, we first consider (1.1) and establish a local existence result. In addition, using the ideas of the "potential well" theory introduced by Payne and Sattinger [28], we show that for some appropriate assumption on g (but without exponential decay property) and the initial data, that this solution is global with energy which decays exponentially under the potential well. Secondly, not only finite time blow up for solutions starting in the unstable set is proved, but also under some appropriate assumptions on g and the initial data, a blow-up result with positive initial energy is established. Finally, we also prove the boundedness of global solutions u(t) to problem (1.1) for strong (ω > 0) damping, namely, u L + ; H 0 1 ( Ω ) W 1 , + ; L 2 ( Ω ) .

This article is organized as follows. In Section 2 we introduce some notation and prepare some material. Section 3 is devoted to global existence for solutions under the potential well and the decay result. In Section 4 we will show that there are solutions of (1.1) with positive initial energy or with arbitrary positive initial energy that blow up in finite time. The last Section we will prove the boundedness of global solutions u(t) to problem (1.1) for strong (ω > 0) damping.

2. Preliminaries

We denote by · q the L q (Ω) norm for 1 ≤ q ≤ ∞ and by · 2 the Dirichlet norm in H 0 1 ( Ω ) . Moreover, for later use we denote by 〈·,·〉 the duality pairing between H-1(Ω) and H 0 1 ( Ω ) . When ω > 0 (resp. ω = 0) for v, w H 0 1 ( Ω ) (resp. for all v, w L2(Ω)), we put
( v , w ) * = ω Ω v w + μ Ω v w , v * = ( v , v ) * 1 / 2 ,

by (1.2), · * is an equivalent norm over H 0 1 ( Ω ) (resp. L2(Ω)).

Let a > 0. Define J a , I a : H 0 1 ( Ω ) by
J a = a 2 u 2 2 - 1 p u p p , I a = a u 2 2 - u p p .
In this case, the "potential depth" is defined as
d a = inf w H 0 1 ( Ω ) \ { 0 } max λ 0 J a ( λ u ) .
It is easy to see that the "potential well" is positive, see [28, 29] for details. Next, we define stable and unstable sets respectively:
W a = u H 0 1 ( Ω ) | I a ( u ) > 0 , J a ( u ) < d a { 0 } , V a = u H 0 1 ( Ω ) | I a ( u ) < 0 , J a ( u ) < d a .
Finally, we consider the energy functional E(t) = E(u(t),u t (t)) defined by
E ( t ) = 1 2 u t ( t ) 2 2 + 1 2 1 - 0 t g ( s ) d s u ( t ) 2 2 + 1 2 g u ( t ) - 1 p u p p ,
where
( g v ) ( t ) = 0 t g ( t - τ ) v ( t ) - v ( τ ) 2 2 d τ .

For the relaxation function g(t) we assume

(G1) g C1[0, ∞) is a non-negative and non-increasing function satisfying
1 - 0 g ( s ) d s = 1 - κ = l > 0 .

( G 2 ) 0 g ( s ) d s < p / 2 - 1 p / 2 - 1 + 1 / ( 2 p ) .

Remark 2.1. Condition (1.3) is needed to establish the local existence result. In fact under this condition, the nonlinearity is Lipschitz from H1(Ω) to L2(Ω). Condition (G1) is necessary to guarantee the hyperbolicity and well-posedness of problem (1.1).

3. Global existence and exponential energy decay

In this section we study the global existence of solutions for problem (1.1). For this purpose, we first consider a related linear problem. Then, we use the well-known contraction mapping theorem to prove the existence of solutions to the nonlinear problem. Throughout the section, we restrict ourselves to the case ω > 0, μ ≠ 0 and n ≥ 3, the other cases being similar (and simpler).

For a given T > 0, we consider the space = C [ 0 , T ] ; H 0 1 ( Ω ) C 1 [ 0 , T ] ; L 2 ( Ω ) equipped with the norm
u 2 = max 0 t T l u ( t ) 2 2 + u t ( t ) 2 2 .
Lemma 3.1. Assume (G 1), (1.2) and (1.3) hold. For every T > 0, every u and every initial data ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) there exists a unique
v C 2 [ 0 , T ] ; H - 1 ( Ω ) such that v t L 2 [ 0 , T ] ; H 0 1 ( Ω ) ,
which solves the linear problem
v t t - Δ v + 0 t g ( t - τ ) Δ v ( τ ) d τ - ω Δ v t + μ v t = u p - 2 u , ( x , t ) Ω × [ 0 , T ] , v ( x , t ) = 0 , ( x , t ) Ω × [ 0 , T ] , v ( x , 0 ) = u 0 ( x ) , v t ( x , 0 ) = u 1 ( x ) x Ω .
(3.1)

Proof. The proof follows from a directly application of the Galerkin method as in [22, 30], thus we omit it here.

Theorem 3.2. Assume (G 1), (1.2) and (1.3) hold. For any initial data ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) , there exists a real number T m > 0 such that problem (1.1) has a unique local weak solution
u C 2 0 , T m ; H - 1 ( Ω ) such that u t L 2 [ 0 , T ] ; H 0 1 ( Ω ) .
If T m < ∞, then
lim t T m l u ( t ) 2 2 + u t ( t ) 2 2 = .
(3.2)
Proof. Taking ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) and letting R = 2 u 0 2 2 + u 1 2 2 . For any T > 0, we consider
= u : u ( 0 ) = u 0 , u t ( 0 ) = u 1 and u R .
By Lemma 3.1, for any u we may define v = Φ(u), being the unique solution to problem (3.1). We claim that, for a suitable T > 0, Φ is a contractive map from into itself. Given u , multiplying (3.1) by v t and integrating over [0,t] [0,T], we have
v t ( t ) 2 2 + 1 - 0 t g ( s ) d s v ( t ) 2 2 + g v ( t ) + 2 0 t v t ( τ ) * 2 d τ = 0 t g v d τ - 0 t g ( τ ) v ( τ ) 2 2 d τ + u 0 2 2 + u 1 2 2 + 2 0 t Ω u ( τ ) p - 2 u ( τ ) v t ( τ ) d x d τ u 0 2 2 + u 1 2 2 + 2 0 t Ω u ( τ ) p - 2 u ( τ ) v t ( τ ) d x d τ ,
(3.3)
here taking into account the condition (G 1). For the last term, using Hölder, Sobolev, and Young inequalities, we have
2 0 t Ω u ( τ ) p - 2 u ( τ ) v t ( τ ) d x d τ C 0 T u ( τ ) 2 * p - 1 v t ( τ ) 2 * d τ C 1 0 T u ( τ ) * p - 1 v t ( τ ) * d τ C 2 T R 2 ( p - 1 ) + 2 0 T v t ( τ ) * 2 d τ ,
(3.4)
where 2* = 2n/(n-2). Combining (3.3) with (3.4) and taking the maximum over [0, T], we get
v 2 1 2 R 2 + C 2 T R 2 ( p - 1 ) .

Choosing T sufficiently small such that C2TR2(p-1)R2/2, we get v R , which shows that Φ maps into itself.

Next, we verify that Φ is a contraction. Taking w1 and w2 in , subtracting the two equations (3.1) for v1 = Φ(w1) and v2 = Φ(w2) and setting w = v1- v2, then we have for all φ H 0 1 ( Ω ) and a.e. t [0,T]
w t t , φ + Ω w ( t ) φ d x + Ω 0 t g ( t - τ ) Δ w ( τ ) d τ φ d x + Ω w t ( t ) φ d x + μ Ω w t ( t ) φ d x = Ω w 1 ( t ) p - 2 w 1 ( t ) - w 2 ( t ) p - 2 w 2 ( t ) φ d x .
(3.5)
By taking φ = w t in (3.5) and arguing as above, we obtain
Φ ( w 1 ) - Φ ( w 2 ) 2 = w 2 C 3 R 2 p - 4 T w 1 - w 2 2 ε w 1 - w 2 2

for some ε < 1 provided T is sufficiently small. This proves the claim. By the contraction mapping principle, there exists a unique (weak) solution to (1.1) defined on [0,T m ).

By the construction above, we observe that the local existence time of u merely depends (through R) on the norms of the initial data. Therefore, as long as u ( t ) remains bounded, the solution may be continued, see also [[31], p. 158], for a similar argument. Hence, if T m < ∞, we have
lim t T m l u ( t ) 2 2 + u t ( t ) 2 2 = lim t T m u ( t ) = .

Before we state and prove our global existence result, we need the following lemmas.

Lemma 3.3. [24, Lemma 2.1] Assume (G 1), (1.2) and (1.3) hold. Let u(t) be a solution of (1.1). Then E(t) is nonincreasing, that is
E ( t ) = - u t ( t ) * 2 + 1 2 ( g u ) ( t ) - 1 2 g ( t ) u ( t ) 2 2 0 .
(3.6)
Moreover, the following energy inequality holds:
E ( t ) + s t u t ( τ ) * 2 d τ E ( s ) , for 0 s t < T m .
(3.7)
Lemma 3.4. Assume (G 1), (1.2) and (1.3) hold, and 0 < al. Let u(x, t) be a local solution of problem (1.1) with initial data ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) . Then the following assertions hold.
  1. (1)

    If there exists a number t0 [0,T m ) such that u(·,t0) W a and E(t0) < d a , then u(·, t) W a and E(t) < d a for all t [t0,T m ).

     
  2. (2)

    If there exists a number t0 [0,T m ) such that u(·, t0) V a and E(t0) < d a , then u(·,t) V a and E(t) < d a for all t [t0,T m ).

     

Proof. The proof is almost the same that of Tsutsumi [32].

The following integral inequality plays an important role in our proof of the energy decay of the solutions to problem (1.1).

Lemma 3.5. [33] Assume that the function φ : + {0} → + {0} is a non-increasing function and that there exists a constant c > 0 such that
t φ ( s ) d s c φ ( t )
for every t [0, ∞). Then
φ ( t ) φ ( 0 ) exp ( 1 - t / c )

for every tc.

Theorem 3.6. Assume (G 1), (1.2) and (1.3) hold, and 0 < al. Let u(x, t) be a local solution of problem (1.1) with initial data ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) . In addition assume that u(0) W a and E(0) < d a , then the corresponding solution to (1.1) globally exists, i.e., T m = ∞. Moreover, if d a < θ and σ = 1 - l > 0 is small sufficiently such that
1 - C p ( Ω ) 2 p E ( 0 ) p - 2 p - 2 2 l - p 2 - 5 p ( 1 - l ) 2 ( p - 2 ) l > 0
where θ = ( p - 2 ) / ( 2 p ) l p / ( p - 2 ) C - 2 p / ( p - 2 ) ( Ω ) and C(Ω) is the optimal constant of Sobolev imbedding H 0 1 ( Ω ) L p ( Ω ) , then the energy decay is
E ( t ) E ( 0 ) exp ( 1 - C - 1 t )

for every t [0, ∞), where C is some positive constant.

Proof. We only consider the case ω > 0 and μ > - λω. In order to get T m = ∞, by Theorem 3.2, it suffices to show that
l u ( t ) 2 2 + u t ( t ) 2 2
is bounded independently of t. Since u(0) W a and E(0) < d a , it follows from Lemma 3.4 that
u ( t , ) W a , E ( t ) < d a for 0 , T m .
On the other hand, since u(t, ·) W a means
J a ( u ) ( p - 2 ) a 2 p u ( t ) 2 2 for 0 , T m .
(3.8)
So, it follows from (3.8) and Lemma 3.3 with s = 0 that
( p - 2 ) a 2 p l u ( t ) 2 2 + 1 2 u t ( t ) 2 2 J a ( u ) + 1 2 u t ( t ) 2 2 E ( t ) + 0 t u t ( τ ) * 2 d τ E ( 0 ) < d a for 0 , T m ,
which implies
l u ( t ) 2 2 + u t ( t ) 2 2 C d a ,

where C is a positive constant depending only on l and p.

From Lemma 3.3 we have
E ( 0 ) E ( t ) 1 2 1 - 0 t g ( s ) d s u ( t ) 2 2 - 1 p u p p l 2 u ( t ) 2 2 - 1 p u p p ,
which together with u(t, ·) W a yields
u ( t ) 2 2 2 p ( p - 2 ) l E ( t ) 2 p ( p - 2 ) l E ( 0 ) .
(3.9)
In addition,
1 - 2 p u ( t ) p p 1 - 2 p C p ( Ω ) u ( t ) 2 p 2 l C p ( Ω ) 2 p ( p - 2 ) l E ( 0 ) p - 2 2 E ( t ) 2 ( 1 - ε ) E ( t ) ,
(3.10)
where
ε = 1 - C p ( Ω ) 2 p ( p - 2 ) E ( 0 ) p - 2 2 l - p 2 .

Note that E(0) < θ, we see that ϵ > 0.

Multiplying (1.1) by u(t) and integrating over Ω × [t1,t2] (0 ≤ t1t2), we get
0 = Ω t 1 t 2 u u t t - Δ u + 0 t g ( t - τ ) Δ u ( τ ) d τ - ω Δ u t + μ u t - u p - 2 u d t d x = Ω u ( t ) u t ( t ) d x t 1 t 2 - t 1 t 2 u t ( t ) 2 2 d t + t 1 t 2 u ( t ) 2 2 d t - t 1 t 2 u ( t ) p p d t + t 1 t 2 ( u ( t ) , u t ( t ) ) * d t + t 1 t 2 Ω 0 t g ( t - τ ) Δ u ( τ ) u ( t ) d τ d x d t = Ω u ( t ) u t ( t ) d x t 1 t 2 - 2 t 1 t 2 u t ( t ) 2 2 d t + 2 t 1 t 2 E ( t ) d t + 2 p - 1 t 1 t 2 u ( t ) p p d t - t 1 t 2 g u ( t ) d t + t 1 t 2 0 t g ( τ ) d τ u ( t ) 2 2 d t + t 1 t 2 ( u ( t ) , u t ( t ) ) * d t + t 1 t 2 Ω 0 t g ( t - τ ) Δ u ( τ ) u ( t ) d τ d x d t .
(3.11)
For the last term in (3.11), one has
- 2 Ω 0 t g ( t - τ ) Δ u ( τ ) u ( τ ) d τ d x = 2 Ω 0 t g ( t - τ ) u ( τ ) u ( t ) d τ d x = 0 t g ( t - τ ) u ( t ) 2 2 + u ( τ ) 2 2 d τ - 0 t g ( t - τ ) u ( t ) - u ( t ) 2 2 d τ .
(3.12)
Combining (3.11) and (3.12), we have
2 t 1 t 2 E ( t ) d t + 2 p - 1 t 1 t 2 u ( t ) p p d t = - Ω u ( t ) u t ( t ) d x t 1 t 2 + 2 t 1 t 2 u t ( t ) 2 2 d t + 1 2 t 1 t 2 g u ( t ) d t - 1 2 t 1 t 2 0 t g ( τ ) d τ u ( t ) 2 2 d t + 1 2 t 1 t 2 0 t g ( t - τ ) u ( τ ) 2 2 d τ d t - t 1 t 2 ( u ( t ) , u t ( t ) ) * d t - Ω u ( t ) u t ( t ) d x t 1 t 2 + 2 t 1 t 2 u t ( t ) 2 2 d t + 1 2 t 1 t 2 ( g u ) ( t ) d t + 1 2 t 1 t 2 0 t g ( t - τ ) u ( τ ) 2 2 d τ d t - t 1 t 2 ( u ( t ) , u t ( t ) ) * d t ,
(3.13)
where the last inequality comes from (G 1). For the left-hand side of the (3.13), by (3.10) we obtain
2 t 1 t 2 E ( t ) d t + 2 p - 1 t 1 t 2 u ( t ) p p d t 2 ε t 1 t 2 E ( t ) d t .
(3.14)
We next estimate every term of the right-hand side of the (3.13). Firstly, by Hölder inequality and Poincaré inequality
Ω u ( t ) u t ( t ) d x 1 2 u ( t ) 2 2 + 1 2 u t ( t ) 2 2 λ 2 u ( t ) 2 2 + E ( t ) .
Using (3.9) we see that
Ω u ( t ) u t ( t ) d x c 1 E ( t ) ,
where c1 is a constant independent on u, from which follows that
Ω u ( t ) u t ( t ) d x t 1 t 2 2 c 1 E ( t 1 ) .
(3.15)
Since u(t, ·) W a , we have 0 < I a (u) ≤ E(t). Thus, from (3.7), we deduce that
t 1 t 2 u t ( t ) * 2 E ( t 1 ) ,
which implies
t 1 t 2 u t ( t ) 2 2 c 2 E ( t 1 ) .
Hence, by Poincaré inequality we get
2 t 1 t 2 u t ( t ) 2 2 d t 2 c 3 E ( t 1 ) ,
(3.16)
where c3 is a constant independent on u. In addition, using Young's inequality for convolution f*g q f r g s with 1/q = 1/r + 1/s - 1 and 1 ≤ q,r,s ≤ ∞, noting that if q = 1, then r = 1 and s = 1, we have
t 1 t 2 0 t g ( t - τ ) u ( τ ) 2 2 d τ d t t 1 t 2 g ( t ) d t t 1 t 2 u ( t ) 2 2 d t ( 1 - l ) t 1 t 2 u ( t ) 2 2 d t .
Further, by (3.9) we then have
t 1 t 2 0 t g ( t - τ ) u ( τ ) 2 2 d τ d t 2 p ( 1 - l ) ( p - 2 ) l t 1 t 2 E ( t ) d t
(3.17)
and
t 1 t 2 0 t g ( t - τ ) u ( τ ) 2 2 d τ d t 2 p ( 1 - l ) ( p - 2 ) l t 1 t 2 E ( t ) d t .
(3.18)
Combining (3.17) and (3.18), we get
1 2 t 1 t 2 ( g u ) ( t ) d t t 1 t 2 0 t g ( t - τ ) u ( t ) 2 2 + u ( τ ) 2 2 d τ d t 4 p ( 1 - l ) ( p - 2 ) l t 1 t 2 E ( t ) d t .
(3.19)
By Poincaré inequality and (3.9), we also have the following estimate
- 2 t 1 t 2 ( u ( t ) , u t ( t ) ) * d t = - t 1 t 2 d d t u ( t ) * 2 = u ( t 1 ) * 2 - u ( t 2 ) * 2 2 λ ω + μ λ u ( t 1 ) 2 2 4 p ( λ ω + μ ) λ ( p - 1 ) l E ( t 1 ) c 4 E ( t 1 ) ,
(3.20)

where c4 is a constant independent on u.

Combining (3.13)-(3.20), we obtain
2 ε t 1 t 2 E ( t ) d t 2 C E ( t 1 ) + 5 p ( 1 - l ) ( p - 2 ) l t 1 t 2 E ( t ) d t ,
where C is a constant independent on u, that is
1 - C p ( Ω ) 2 p E ( 0 ) p - 2 p - 2 2 l - p 2 - 5 p ( 1 - l ) 2 ( p - 2 ) l t 1 t 2 E ( t ) d t C E ( t 1 ) .
(3.21)
Denote
a = 1 - C p ( Ω ) 2 p E ( 0 ) p - 2 p - 2 2 l - p 2 - 5 p ( 1 - l ) 2 ( p - 1 ) l .
We rewrite (3.21)
a t E ( τ ) d τ C E ( t )

for every t [0, ∞).

Since a > 0 when σ = 1 - l > 0 small sufficiently by Lemma 3.5, we obtain the following energy decay for problem (1.1) as
E ( t ) < E ( 0 ) exp ( 1 - a C - 1 t )

for every tCa-1.

Remark 3.1. For the definition of d a and Sobolev imbedding inequality, we have
d a p - 2 2 p a p p - 2 u 2 u p 2 p p - 2
and
u p C u 2 .

Since L p ( Ω ) H 0 1 ( Ω ) is compact, the best constants and the best function v(x) in the above Sobolev imbedding inequality can be attained. For example, n = 1, p = ∞, Ω = (c, d) , the best C and the best function v(x) are attained, see [34]. In this case, d a p - 2 2 p a p p - 2 C - 2 p p - 2 < θ . Then, we can take the initial data u0 = v(x) which yields the set of the initial data that yields the exponential decay is not empty.

4. Blow-up solution

In this section, we deal with the blow-up solutions of problem (1.1). The basic idea comes from [30], however our argument contains nontrival modifications.

Lemma 4.1. Assume (G 1), (1.2) and (1.3) hold. Let u(x,t) be a local solution of problem (1.1) on [0,T m ) with initial data ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) . If T m < ∞, then
lim t T m u ( t ) q = for all q 1 such that n ( p - 2 ) 2 < q < p .
(4.1)

Moreover, if n ≥ 3 and p = 2n/(n - 2) = 2* (ω > 0), then (4.1) also holds for q = p.

Proof. From (3.7), we have
1 2 u t ( t ) 2 2 + l 2 u ( t ) 2 2 1 p u ( t ) p p + E ( 0 ) , t 0 , T m ,
(4.2)
which, together with (3.2), implies
lim t T m u ( t ) p = .
(4.3)
This proves (4.1) at once for the case of p = q = 2n/(n - 2). For the remaining cases, notice (4.3) that implies
lim t T m u ( t ) 2 = .
(4.4)
Moveover, by (4.2) we obtain
l u ( t ) 2 2 2 p u ( t ) p p + 2 E ( 0 ) , t 0 , T m .
From the Gagliardo-Nirenberg inequality we have
u ( t ) p p C u ( t ) q p ( 1 - α ) u ( t ) 2 p α for α = 2 n ( p - q ) p ( 2 n + 2 q - n q ) ,
which yields
u ( t ) 2 2 2 l E ( 0 ) + C 1 C u ( t ) q p ( 1 - α ) u ( t ) 2 p α .

Since n(p - 2)/2 < q < p implies 0 < α < 1 and < 2, the above inequality combined with (4.4) immediately yields (4.1).

Next we will prove the main blow-up result by the concavity method of Levine [35, 36] and the estimates similar as [30].

Theorem 4.2. Assume (G 1), (G 2), (1.2) and (1.3) hold. Let u(x,t) be a local solution of problem (1.1) with initial data ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) . If ω > 0, then there is a real number t0 [0,T m ) such that u(t0, ·) V k and E(t0) < d k if and only if T m < ∞, where
k = l - 1 p ( p - 2 ) 0 g ( s ) d s .
Proof. We first consider "if part", without loss of generality, we may assume that t0 = 0. Assume by contradiction that the solution u is global. Then, for any T > 0 we consider H(t) : [0,T] → + defined by
H ( t ) = u ( t ) 2 2 + 0 t u ( τ ) * 2 d τ + ( T - t ) u 0 * 2 .
A direct computation yields
H ( t ) = 2 Ω u ( t ) u t ( t ) d x + u ( t ) * 2 - u 0 * 2 = 2 Ω u ( t ) u t ( t ) d x + 2 0 t ( u ( τ ) , u t ( τ ) ) * d τ
and
H ( t ) = 2 u t t , u ( t ) + 2 u t ( t ) 2 2 + 2 ( u ( t ) , u t ( t ) ) * , for a .e . t [ 0 , T ] .
By multiplying (1.1) by u and integrating over Ω, we have
u t t , u ( t ) + ( u ( t ) , u t ( t ) ) * = - u ( t ) 2 2 - Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x + u ( t ) p p ,
(4.5)
which implies
H ( t ) = 2 u t ( t ) 2 2 - 2 u ( t ) 2 2 - 2 Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x + 2 u ( t ) p p .
Therefore, we have
H ( t ) H ( t ) - p + 2 4 H ( t ) 2 = 2 H ( t ) u t ( t ) 2 2 - u ( t ) 2 2 - Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x + u ( t ) p p - ( p + 2 ) Ω u ( t ) u t ( t ) d x + 0 t u ( τ ) , u t ( τ ) * d τ 2 = 2 H ( t ) u t ( t ) 2 2 - u ( t ) 2 2 - Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x + u ( t ) p p + ( p + 2 ) G ( t ) - H ( t ) - ( T - t ) u 0 * 2 u t ( t ) 2 2 + 0 t u t ( τ ) * 2 d τ ,
(4.6)
where G(t) : [0,T] → + is the function defined by
G ( t ) = u ( t ) 2 2 + 0 t u ( τ ) * 2 d τ u t ( t ) 2 2 + 0 t u t ( τ ) * 2 d τ - Ω u ( t ) u t ( t ) d x + 0 t ( u ( τ ) , u t ( τ ) ) * d τ 2 .
Using the Schwarz inequality, we have
Ω u ( t ) u t ( t ) d x 2 u ( t ) 2 2 u t ( t ) 2 2 , 0 t ( u ( τ ) , u t ( τ ) ) * d τ 2 0 t u ( τ ) * 2 d τ 0 t u t ( τ ) * 2 d τ ,
and
Ω u ( t ) u t ( t ) d x 0 t ( u ( τ ) , u t ( τ ) ) * d τ u ( t ) 2 0 t u t ( τ ) * 2 d τ 1 2 u t ( t ) 2 0 t u ( τ ) * 2 d τ 1 2 1 2 u ( t ) 2 2 0 t u t ( τ ) * 2 d τ + 1 2 u t ( t ) 2 2 0 t u ( τ ) * 2 d τ .
These three inequalities entail G(t) ≥ 0 for every [0, T]. Using (4.6), we get
H ( t ) H ( t ) - p + 2 4 H ( t ) 2 H ( t ) L ( t ) for a .e . t [ 0 , T ] ,
(4.7)
where
L ( t ) = - p u t ( t ) 2 2 - 2 u ( t ) 2 2 + 2 u ( t ) p p - ( p + 2 ) 0 t u t ( t ) * 2 d τ - 2 Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x .
(4.8)
For the last term on the left of (4.8), we have
- Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x = 0 t g ( t - τ ) Ω u ( τ ) u ( τ ) d x d τ = 0 t g ( t - τ ) Ω u ( t ) u ( τ ) - u ( τ ) d x d τ + 0 t g ( t - τ ) u ( t ) 2 2 d τ = 0 t g ( t - τ ) Ω u ( t ) ( u ( τ ) - u ( τ ) ) d x d τ + 0 t g ( τ ) u ( t ) 2 2 d τ .
(4.9)
Combining (4.8) with (4.9), we get
L ( t ) = - p u t ( t ) 2 2 - 2 1 - 0 t g ( τ ) d τ u ( t ) 2 2 + 2 u ( t ) p p + 2 0 t g ( t - τ ) Ω u ( t ) ( u ( τ ) - u ( τ ) ) d x d τ - ( p + 2 ) 0 t u t ( τ ) * 2 d τ - p u t ( t ) 2 2 - 2 1 - 0 t g ( τ ) d τ u ( t ) 2 2 + 2 u ( t ) p p - 2 p 2 0 t g ( t - τ ) | u ( τ ) - u ( t ) | 2 d τ + 1 2 p 0 t g ( τ ) u ( t ) 2 2 d τ - ( p + 2 ) 0 t u t ( τ ) * 2 d τ - 2 p E ( t ) + ( p - 2 ) 1 - 0 t g ( τ ) d τ u ( t ) 2 2 - 1 p 0 t g ( τ ) u ( t ) 2 2 d τ - ( p + 2 ) 0 t u t ( τ ) * 2 d τ .
Using (3.7), we have
E ( t ) + 0 t u t ( τ ) * 2 d τ E ( 0 ) ,
and then
L ( t ) - 2 p E ( 0 ) + ( p - 2 ) 1 - 0 t g ( τ ) d τ u ( t ) 2 2 - 1 p 0 t g ( τ ) u ( t ) 2 2 d τ + ( p - 2 ) 0 t u t ( τ ) * 2 d τ - 2 p E ( 0 ) + ( p - 2 ) 0 t u t ( τ ) * 2 d τ + ( p - 2 ) 1 - 0 t g ( τ ) d τ - 1 p 0 t g ( τ ) d τ u ( t ) 2 2 2 p p - 2 2 p 1 - 0 t g ( τ ) d τ - 1 p ( p - 2 ) 0 t g ( τ ) d τ u ( t ) 2 2 - E ( 0 ) + ( p - 2 ) 0 t u t ( τ ) * 2 d τ 2 p p - 2 2 p l - 1 p ( p - 2 ) 0 t g ( τ ) d τ u ( t ) 2 2 - E ( 0 ) + ( p - 2 ) 0 t u t ( τ ) * 2 d τ 2 p p - 2 2 p k u ( t ) 2 2 - E ( 0 ) + ( p - 2 ) 0 t u t ( τ ) * 2 d τ .
Since
0 g ( s ) d s < p / 2 - 1 p / 2 - 1 + 1 / ( 2 p ) ,
we have
0 < k = l - 1 p ( p - 2 ) 0 g ( τ ) d τ l .
By Lemma 3.4, we have
u ( t , ) V k and E ( t ) < d k for t 0 , T m .
Then, we have
d k p - 2 2 p k p p - 2 u ( t ) 2 2 p p - 2 u ( t ) p 2 p p - 2 p - 2 2 p k u ( t ) 2 2 , for t 0 , T m .
(4.10)
The above inequality comes from [29]; see [28, 29] for further details. Since E(0) < d k , there exists δ > 0 (independent of T) such that
L ( t ) δ , for t [ 0 , T ] .
(4.11)
From (4.10) and the definition of H(t), there also exists ρ > 0 (independent of T) such that
H ( t ) ρ , for t [ 0 , T ] .
(4.12)
By (4.7), (4.11), and (4.12) it follows that
H ( t ) H ( t ) - p + 2 4 H ( t ) 2 δ ρ , for a .e . t [ 0 , T ] .
Setting y(t) = H(t)-(p-2)/4, then we have
y ( t ) p - 2 4 δ ρ y ( t ) p + 6 p - 2 , for a .e . t [ 0 , T ] ,
which implies that y(t) reaches 0 in finite time, say as tT*. Since T* is independent of the initial choice of T, we may assume that T* < T. This tells us that
lim t T * H ( t ) = .
In turn, this implies that
lim t T * u ( t ) 2 2 = .
(4.13)
Indeed, if u ( t ) 2 2 as tT*, then (4.13) immediately follows. On the contrary, if u ( t ) 2 2 remains bounded on [0,T*), then
lim t T * 0 t u ( τ ) * 2 d τ = ,

so that again (4.13) is satisfied. This implies a contradiction, i,e., T m < ∞.

Conversely, for "only if part" we assume now that T m < ∞. Notice first that, for every t > 0, there holds
0 t u t ( τ ) * 2 d τ 1 t 0 t u t ( τ ) * d τ 2 1 t u ( t ) * - u * 2 .
Hence, by (3.7) and 0 < kl, we have
1 2 I k ( u ( t ) ) E ( t ) E ( 0 ) - 1 t u ( t ) * - u * 2 .
(4.14)
By Lemma 4.1 we have u(t)2 → ∞ as tT m , i.e., u(t)* → ∞ as tT m , together with (4.14) which implies
lim t T m I k ( u ( t ) ) = lim t T m E ( t ) = - .
(4.15)
Since J k (u(t)) ≤ E(t), by (4.15) we obtain that
J k ( u ( t 0 ) ) E ( t 0 ) < d k I k ( u ( t 0 ) ) < 0

for some t0 [0,T m ). These imply u(t0) V k , E(t0) < d k .

Remark 4.1. The "if part" of Theorem 4.2 means that the solution to (1.1) blows up in a finite time for suitable "large" initial data u0 and u1 in the sense of u0 V k and E(0) < d k . Also, (4.15) is an essential behavior for which the solution of (1.1) blows up in a finite time.

Remark 4.2. In Theorem 4.2, we restrict ω > 0 in order to prove the "only if part". In fact, if ω > 0, it is easy to obtain u(t)* → ∞ as tT m from u(t)2 → ∞ as tT m , which implies E(t) → -∞ as tT m . If ω = 0 (only with weak damping), assuming 2 < p ≤ 2 + 2/n, then we can obtain u (t)2 → ∞ as tT m (see [37] for details) which yields that Theorem 4.2 also holds for the case of ω = 0 with 2 < p ≤ 2 + 2/n.

Next, we consider the blow-up solution of problem (1.1) for the case of weak damping (ω = 0) with arbitrary positive initial energy. We need an addition assumption on the relaxation function g:

(G3) The function of e t 2 g ( t ) is of positive in the following sense:
0 t v ( s ) 0 s e s - r 2 g ( s - τ ) v ( τ ) d τ d s 0 ,

v C1([0,∞)) and t > 0.

Obviously, g(t) = εe-twith 0 < ε < 1 satisfies assumptions (G1)-(G3). Let
I ( u ) = I 1 ( u ) = u 2 2 - u p p .
Lemma 4.3. [27, Lemma 2.1]) Assume that g(t) satisfies (G 1), (G 3) and Λ(t) is a function that is twice continuously differentiable, satisfying
Λ ( t ) + μ Λ ( t ) > 0 t g ( t - τ ) Ω u ( τ ) u ( t ) d x d τ , Λ ( 0 ) > , Λ ( 0 ) > 0 ,

for every t [0,T m ), where u(t) is the corresponding solution of problem (1.1) with weak damping. Then the function Λ(t) is strictly increasing on [0,T m ).

Lemma 4.4. Suppose that u 0 H 0 1 ( Ω ) , u1 L2(Ω) satisfy
Ω u 0 ( x ) u 1 ( x ) d x > 0 .

If the local solution u(t) of problem (1.1) with weak damping exists on [0,T m ) and satisfies I(u(t)) < 0, then u ( t ) 2 2 is strictly increasing on [0,T m ).

Proof. Since u(t) is the local solution of problem (1.1) with weak damping, by a simple computation we have
1 2 d 2 d t 2 u ( t ) 2 2 = Ω u t ( t ) 2 + u u t t d x = u t ( t ) 2 2 - μ Ω u u t d x - I ( u ( t ) ) + 0 t g ( t - τ ) Ω u ( τ ) u ( t ) d x d τ - μ Ω u u t d x + 0 t g ( t - τ ) Ω u ( τ ) u ( t ) d x d τ ,
where the last inequality uses I(u(t)) < 0. Then we get
d 2 d t 2 u ( t ) 2 2 + μ d d t u ( t ) 2 2 > 0 t g ( t - τ ) Ω u ( τ ) u ( t ) d x d τ .

Therefore, this lemma comes from Lemma 4.3.

Theorem 4.5. Assume (G 1), (G 3), (1.2) and (1.3) hold. Let u(x,t) be a local solution of problem (1.1) with initial data ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) . If ω = 0, g(s) also satisfies
0 g ( s ) d s < ( p - 2 ) 2 ( p - 1 ) 2 ,
(4.16)
and (u0,u1) satisfies the following conditions
E ( 0 ) > 0 , Ω u 0 ( x ) u 1 ( x ) d x > 0 , I ( u 0 ) < 0 , u 0 2 2 > 2 p E ( 0 ) ( k p - 2 ) λ .
(4.17)
where
k = l - 1 p ( p - 2 ) 0 g ( s ) d s > 0

and λ is the first eigenvalue of the operatorunder homogeneous Dirichlet boundary conditions, then the corresponding solution u(t) of problem (1.1) blows up in a finite time, i.e., T m < ∞.

Proof. Without loss of generality, we may assume μ = 1. First, by a contradiction argument we claim that
I ( u ( t ) ) < 0 ,
(4.18)
and
u ( t ) 2 2 > 2 p E ( 0 ) ( k p - 2 ) λ ,
(4.19)
for every t [0,T m ). If this was not the case, then there would exist a time t1 such that
t 1 = min t ( 0 , T m ) : I ( u ( t ) ) = 0 > 0 .
(4.20)
By the continuity of the solution u(t) as a function of t, we see that I(u(t)) < 0 when t (0,t1) and I(u(t)) = 0. Thus by Lemma 4.4 we have
u ( t ) 2 2 > u 0 2 2 > 2 p E ( 0 ) ( k p - 2 ) λ ,
for every t [0,t1). In addition, it is obvious that u ( t ) 2 2 is continuous on [0,t1]. Thus the following inequality is obtained:
u ( t 1 ) 2 2 > 2 p E ( 0 ) ( k p - 2 ) λ .
(4.21)
On the other hand, it follows from the definition of E(t) and (3.7) that
1 2 1 - 0 t 1 g ( s ) d s u ( t 1 ) 2 2 - 1 p u ( t ) p p E ( 0 ) .
(4.22)
Since 0 < k l 1 - 0 t 1 g ( s ) d s , from (4.22), we have
k 2 u ( t 1 ) 2 2 - 1 p u ( t ) p p E ( 0 ) .
Noting the fact that I(u(t1)) = 0, we then have
k p - 2 2 p u ( t 1 ) 2 2 E ( 0 ) .
Thus, by the Poincaré inequality and (4.16) we have
u ( t 1 ) 2 2 2 p E ( 0 ) ( k p - 2 ) λ .
(4.23)

Obviously, there is a contradiction between (4.21) and (4.23). Thus, we have proved that (4.18) is true for every for every t [0,T m ). Furthermore, by Lemma 4.4 we see that (4.19) is also valid on t [0,T m ).

Secondly, we prove that the solution of problem (1.1) blows up in a finite time. The proof is similar "if part" in the Theorem 4.2, for the convenience of the readers, we give the sketch of the proof here. Assume by contradiction that the solution u is global. Then, for sufficiently large T > 0 we consider Φ(t) : [0,T] → + defined by
Φ ( t ) = u ( t ) 2 2 + 0 t u ( τ ) 2 2 d τ + ( T - t ) u 0 2 2 .
A direct computation yields
Φ ( t ) = 2 Ω u ( t ) u t ( t ) d x + u ( t ) 2 2 - u 0 2 2 = 2 Ω u ( t ) u t ( t ) d x + 2 0 t ( u ( τ ) , u t ( τ ) ) d τ
and
Φ ( t ) = 2 u t t , u ( t ) + 2 u t ( t ) 2 2 + 2 u ( t ) , u t ( t ) , for a .e . t [ 0 , T ] .
where (u(t),u t (t)) = Ωu(t)u t (t)dx. By multiplying (1.1) by u and integrating over Ω, we have
u t t , u ( t ) + u ( t ) , u t ( t ) = - u ( t ) 2 2 - Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x + u ( t ) p p ,
which implies
Φ ( t ) = 2 u t ( t ) 2 2 - 2 u ( t ) 2 2 - 2 Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x + 2 u ( t ) p p .
Therefore, we have
Φ ( t ) Φ ( t ) - p + 2 4 Φ ( t ) 2 = 2 Φ ( t ) u t ( t ) 2 2 - u ( t ) 2 2 - Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x + u ( t ) p p - p + 2 Ω u ( t ) u t ( t ) d x + 0 t u ( τ ) , u t ( τ ) d τ 2 = 2 Φ ( t ) u t ( t ) 2 2 - u ( t ) 2 2 - Ω 0 t g ( t - τ ) Δ u ( τ ) d τ u ( t ) d x + u ( t ) p p + p + 2 Ψ ( t ) - Φ ( t ) - ( T - t ) u 0 2 2 u t ( t ) 2 2 + 0 t u t ( τ ) 2 2 d τ ,
(4.24)
where Ψ(t) : [0,T] → + is the function defined by
Ψ ( t ) = u ( t ) 2 2 + 0 t u ( τ ) 2 2 d τ u t ( t ) 2 2 + 0 t u t ( τ ) 2 2 d τ - Ω u ( t ) u t ( t ) d x + 0 t u ( τ ) , u t ( τ ) d τ 2 .
Using the Schwarz inequality, we have
Ω u ( t ) u t ( t ) d x 2 u ( t ) 2 2 u t ( t ) 2 2 , 0 t u ( τ ) , u t ( τ ) d τ 2 0 t u ( τ ) 2 2 d τ 0 t u t ( τ ) 2 2 d τ ,
and
Ω u ( t )