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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