Abstract

This paper is concerned with a problem of a logarithmic nonuniform flexible structure with time delay, where the heat flux is given by Cattaneo’s law. We show that the energy of any weak solution blows up infinite time if the initial energy is negative.

1. Introduction

In this work, we consider the vibrations of an inhomogeneous flexible structure system with a constant internal delay and logarithmic nonlinear source term:with boundary conditionsand initial conditionswhere is the displacement of a particle at position , and the time . is the coupling constant depending on the heating effect,,, and are positive constants, and is a real number. is the relaxation time describing the time lag in the response for the temperature, and represents the time delay in particular if reduces to the viscothermoelastic system with delay, in which the heat flux is given by Fourier’s law instead of Cattaneo’s law, where is the heat flux, and , and are responsible for the inhomogeneous structure of the beam and, respectively, denote mass per unit length of structure, coefficient of internal material damping (viscoelastic property), and a positive function related to the stress acting on the body at a point . The model of heat condition, originally due to Cattaneo, is of hyperbolic type. We recall the assumptions of , and in [1, 2] such that

In these kinds of problems, Gorain [3] in 2013 has established uniform exponential stability of the problemwhich describes the vibrations of an inhomogeneous flexible structure with an exterior disturbing force. More recently, Misra et al. [4] showed the exponential stability of the vibrations of a inhomogeneous flexible structure with thermal effect governed by the Fourier law.

In addition, we can cite other works in the same form like the system in [5]; Racke studied the exponential stability in linear and nonlinear 1d of thermoelasticity system with second sound given by

Now for the multidimensional system, Messaoudi in [6] established a local existence and a blow-up result for a multidimensional nonlinear system of thermoelasticity with second sound (see in this regard Refs. [7–10]); for the same problem above, Alves et al. proved that system (7) is polynomial decay (see [1]), with boundary and initial conditions:

We know that the dynamic systems with delay terms have become a significant examination subject in differential condition since the of the only remaining century. The delay effect that is similar to memory processes is important in the research of applied mathematics such as physics, noninstant transmission phenomena, and biological motivation; model (7) is related to the following problem with delay terms:

The authors prove that the system (9) is well posed and exponential decay under a small condition on time delay (see [2]). Now in the presence of source term the system (9) becomes the system studied in this work with a logarithmic source term; this type of problems is encountered in many branches of physics such as nuclear physics, optics, and geophysics. It is well known, from the quantum field theory, that such kind of logarithmic nonlinearity appears naturally in inflation cosmology and in supersymmetric field theories (see [11–13]).

This work is organized as follows: In “Statement of Problem,” we talk briefly about the local existence of the systems (1), (2), and (3), and we define some space and theorem used. In “Blow-up of Solution,” the blow-up result is proved.

2. Statement of Problem

Let us introduce the function

Thus, we have

Then, problems (1)–(3) are equivalent to

We first state a local existence theorem that can be established by combining the arguments of related works ^10,6

Let and denote by

The state space of is the Hilbert space

Theorem 1. Assume thatThen, for every there exists a unique local solution in the class

3. Blow-up of Solution

In this section, we prove that the solutions for the problems (12)–(13) blow up in a finite time when the initial energy is negative. We use the improved method of Salim and Messaoudi [6] We define the energy associated with problems (12)–(13) by

Lemma 2. Suppose thatThen, there exists a positive constant depending on only, such thatfor any and , provided that

Proof. If , thenIf , then we setand, for any we haveWe choose to getCombining (20) and (23), the result was obtained.

Lemma 3. There exists a positive constant depending on only, such thatfor any , provided that

Proof. We setthus

By using the inequalities, we have the following corollary.

Corollary 4. There exists a positive constant depending on only, such thatprovided that

Lemma 5. There exists a positive constant depending on only, such thatfor any and .

Proof. If , thenIf , then Using the Sobolev embedding theorems, we haveNow we are ready to state and prove our main result. For this purpose, we define

Corollary 6. Assume that (18) holds. Thenfor any and

Theorem 7. Assume that (18) holds. Assume further thatThen, the solution of (12) blows up in finite time.

Proof. we haveandHenceConsequently, we getby virtue of (17) and (31). We then introducewhere to be specified later andA direct differentiation of givesusing the inequality of youngandSubstituting (41), (42), (43), (44), (45), and (46) in (40), we getWe obtain from (35) and (47) the following:We also set hence, (48) giveswhere and are strictly positive constants depending only on
For , we haveUsing (27), (37) and Young’s inequality, we findExploiting (39), we haveThus, lemma 1 yieldsCombining (50) and (53), we obtainAt this point, we choose so small thatand so large thatOnce and are fixed, we pick so small so thatHence, (54) becomeswhere are strictly positive constants depending only on
Thus, for some , estimate (58) becomesandNext, using Hôlder’s inequality and the embedding , we haveand exploiting Young’s inequality, we obtainTo be able to use Lemma 5, we take which gives
Therefore, for estimate (62) yieldsHence, Lemma 5 givesand with the same way, we getFrom (64), (65), and (66) we obtainCombining (67) and (59), we arrive atwhere is a positive constant depending only on and .
A simple integration of (68) over yieldsTherefore, blows up in timeThe proof is completed.