The Format of the IJOPCM, first submission

Blow-up result in a Cauchy problem for the nonlinear viscoelastic Petrovsky equation

Erhan Pişkin

Dicle University, Department of Mathematics, 21280 Diyarbakır, Turkey



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.




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





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



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




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




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




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.




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



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


Proof: By multiplying the equation in (5) by  and integrating over  we obtain (7).




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


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


the last term on the left side of (9) can be estimated as follows




Inserting (10) into (9), to get

To apply Lemma 3, we define






Then, eq (5) is used to estimate (13) as follows



On using

Eq. (14) becomes



We then use Young inequality to estimate the second term in (15). Namely,


By combining (15) and (16), we get


From (12), (13) and (17), we obtain


Now, we exploit (6) to substitute for

Thus (18) takes the form


At this point we choose  so that


This is, of course, possible by (8). We then conclude, from (19), that


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


From the definition of  we get


Combining (20)-(21), we have

From assumptions of Theorem, we deduce by continuity that there exists  such that


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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