In this paper, we consider an initial-boundary value problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms. We proved a blow up result for the solution with negative initial energy if p > m, and a global result for p ≤ m.
A purely elastic material has a capacity to store mechanical energy with no dissipation (of the energy). A complete opposite to an elastic material is a purely viscous material. The important thing about viscous materials is that when the force is removed it does not return to its original shape. Materials which are outside the scope of these two theories will be those for which some, but not all, of the work done to deform them can be recovered. Such materials possess a capacity of storage and dissipation of mechanical energy. This is the case for viscoelastic material. The dynamic properties of viscoelastic materials are of great importance and interest as they appear in many applications to natural sciences. Many authors have given attention to this problem for quite a long time, especially in the last two decades, and have made a lot of progress.
In [
where
For the problem (1.1) in
where
energy such that
In the absence of the viscoelastic term
In [
in a bounded domain and
In [
where
In this work, we intend to study the following initial-boundary value problem:
where
for the problem (1.6), the memory term
and we consider the strong damping term
Now, we shall add a new variable
A direct computation yields
Thus, the original memory term can be written as
and we get a new system
with the initial conditions
and boundary conditions
The paper is organized as follows. In Section 2, we first prove the blow up result, and then in Section 3, we prove the global existence result.
For convenience, we denote the norm and scalar product in
In this section, we present some materials needed in the proof of our results, state a local existence result, which can be established, combining the argument of [
(G1)
(G2)
(G3) There exists a constant
We start with a local existence theorem which can be established by the Faedo-Galerkin methods. The interested readers are referred to Cavalcanti, Domingos Cavalcanti and Soriano [
Theorem 2.1. Assume (G1) holds. Let
Then for any initial data
with compact support, problem (1.10) has a unique solution
for some
Lemma 2.2. Assume (G1), (G2), (G3) and (2.1) hold. Let
where
Proof. By multiplying the Equation in (1.10) by
For the fourth term on the left side (2.4), by using (1.11), (G2) and (G3), we have
where
Then, we obtain
So, we have
where
Our main result reads as follows.
Lemma 2.3. Suppose that (2.1) holds. Then there exists a positive constant
for any
Proof. If
So, we obtain
If
Therefore (2.9) follows.
We get
and use, throughout this paper, C to denote a generic positive constant.
As a result of (2.3) and (2.5), we have
Corollary 2.4. Suppose that (2.1) holds. Then, we have
for any
Lemma 2.5. (
where
Proof. We set
By taking a derivative of
If
Then, we have
If
The proof is completed.
Next, we have the following theorem concerning blow up.
Theorem 2.6. Assume (G1), (G2), (G3) and (2.1) hold. Let
if
Proof. From (2.2), we have
consequently, we have
Similar to [
where
By multiplying (1.10) by
By using Holder inequality and Young’s inequality to estimate the fourth term on the right hand side of (2.19)
for some number
Then, we have
that is
By using Holder inequality and Young’s inequality to estimate the last two terms on right hand side of (2.24), we obtain
and
and
Substituting (2.24), (2.25) and (2.26) and to (2.23), we have
by taking
by taking proper
so, we have
From (2.16), we have
Then, hence (2.31) yields
where
Writing
where
From (2.3) and (G1) we have
writing
at this point, we choose
By using Holder inequality and Young’s inequality, we next estimate
and
which implies
where
By using
According to (2.36) and (2.41), we get
where
So, we know
In this section, we show that solution of (1.10) is global if
Lemma 3.1. For
Proof.
so,
Theorem 3.2. Assume (G1), (G2) and (G3) hold. Let
for any
Proof. Similar to [
from (2.3), we have
By differentiating
By using Holder inequality and Young’s inequality, we next estimate
Setting
Substituting (3.5) to (3.3), we have
so, there exists a small enough constant
Then, by using Gronwall inequality and continuation principle, we complete the proof of the global existence result.
The authors express their sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. These contributions greatly improved the paper.
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11161057