Blow-up result in a Cauchy problem for the nonlinear viscoelastic Petrovsky equation |
Erhan Pişkin |
Dicle University, Department of Mathematics, 21280 Diyarbakır, Turkey E-mail: episkin@dicle.edu.tr |
Copyright © 2014 Erhan Pişkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, we consider a Cauchy problem for the nonlinear viscoelastic Petrovsky equation. We obtain the blow up of solutions by applying a lemma due to Zhou.
Keywords: Blow Up; Cauchy Problem; Nonlinear Viscoelastic Petrovsky Equation.
1. INTRODUCTION
In [5], Li et al. considered the following nonlinear viscoelastic Petrovsky problem
(1) |
where is a bounded domain in with a smooth boundary is the unit outer normal on and g is a nonnegative memory term. They established some asymptotic behavior and blow up results for solutions with positive initial energy.
Guesmia [3] studied the problem
(2)
where is a bounded function? Under some assumptions, he showed the solution of (2) decay results by using the semigroup method. In [7], Messaoudi investigated the semilinear Petrovsky equation
(3)
He showed that the solution blows up in finite time if and while it exists globally if in [9], Wu and Tsai showed that the solution of (3) is global under some conditions. Also, Chen and Zhou [2] studied the blow up of the solution of (3).
Recently, Li et al. [6] considered the following Petrovsky equation
(4)
The authors obtained global existence, decay and blow up of the solution. Very recently, Pişkin and Polat [8] studied the decay of the solution of the problem (4).
In this paper, our aim is to extend the result of [5], established in bounded domains, to the problem in unbounded domains. Namely, we consider the following Cauchy problem
(5)
where are functions to be specified later.
This paper is organized as follows. In section 2, we present some notations, lemmas, and the local existence theorem. In section 3, under suitable conditions on the initial data, we prove a finite time blow up result.
2. PRELIMINARY NOTES
In this section, we give some assumptions and lemmas which will be used throughout this work. Hereafter we denote by and the norm of and respectively. First, we make the following assumptions
(G) is a nonincreasing differentiable function such that
Next, we state the local existence theorem of the problem (5), which can be established by combining the arguments of [1], [7].
Theorem 1: (Local existence). Suppose that (G) holds, and if and if Then for any initial data with compact support, the problem (5) has a unique local solution
for small enough.
To obtain the result of this paper, we will introduce the modified energy functional
(6)
where
The next lemma shows that our energy functional (6) is a nonincreasing function along the solution of (5).
Lemma 2: is a nonincreasing function for and
(7)
Proof: By multiplying the equation in (5) by and integrating over we obtain (7).
3. BLOW UP OF SOLUTIONS
In this section, we shall show that the solution of the problem (5) blow up in finite time, by the similar arguments as in [4]. For the purpose, we give the lemma.
Lemma 3: [10] Suppose that is a twice continuously differentiable function satisfying
where are constants. Then, blow up in finite time.
Theorem 4: Suppose that (G) holds, and if and if Assume further that
(8)
Then for any initial data with compact support, satisfying
Then the corresponding solution blows up in finite time. In other words, there exists a positive constant such that
Proof: By multiplying the equation in (5) by and integrating over using integrating by parts, we obtain
(9)
the last term on the left side of (9) can be estimated as follows
(10)
Inserting (10) into (9), to get
To apply Lemma 3, we define
(11)
Therefore
(12)
and
(13)
Then, eq (5) is used to estimate (13) as follows
(14)
On using
Eq. (14) becomes
(15)
We then use Young inequality to estimate the second term in (15). Namely,
(16)
By combining (15) and (16), we get
(17)
From (12), (13) and (17), we obtain
(18)
Now, we exploit (6) to substitute for
Thus (18) takes the form
(19)
At this point we choose so that
and
This is, of course, possible by (8). We then conclude, from (19), that
(20)
Now, we use Hölder inequality to estimate as follows
where is such that
and is the ball, with radius centered at the origin. If we call the volume of the unit ball then
(21)
From the definition of we get
(22)
Combining (20)-(21), we have
From assumptions of Theorem, we deduce by continuity that there exists such that
so
Consequently, (22) implies that
It is easy to verify that the requirements of Lemma 3 are satisfied by
Therefore blow up in finite.
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