A New Result for a Blow-up of Solutions to a Logarithmic Flexible Structure with Second Sound

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


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 conditions and initial conditions where uðx, tÞ is the displacement of a particle at position x ∈ ½0, L, and the time t > 0. η > 0 is the coupling constant depending on the heating effect,p ≥ 2,γ, β, and k are positive constants, and μ is a real number. τ > 0 is the relaxation time describing the time lag in the response for the temperature, and τ 0 > 0 represents the time delay in particular if τ = 0ð 1:1Þ reduces to the viscothermoelastic system with delay, in which the heat flux is given by Fourier's law instead of Cattaneo's law, where q = qðx, tÞ is the heat flux, and mðxÞ, δðxÞ, and pðxÞ 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 x. The model of heat condition, originally due to Cattaneo, is of hyperbolic type. We recall the assumptions of mðxÞ, δðxÞ, and pðxÞ in [1,2] such that In these kinds of problems, Gorain [3] in 2013 has established uniform exponential stability of the problem 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][8][9][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 1970s 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 natu-rally in inflation cosmology and in supersymmetric field theories (see [11][12][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.

Statement of Problem
Let us introduce the function Thus, we have ð11Þ We first state a local existence theorem that can be established by combining the arguments of related works 10,6 : Let v = u t and denote by The state space of Φ is the Hilbert space Theorem 1. Assume that Then, for every Φ 0 ∈ , there exists a unique local solution in the class Φ ∈ Cð½0, T,Þ: 2 Advances in Mathematical Physics

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 that Then, there exists a positive constant C > 0 depending on ½0:L only, such that for any u ∈ H 1 0 ð0, LÞ and 2 ≤ s ≤ p, provided that If Ð L 0 juj p ln juj γ dx ≤ 1, then we set and, for anyβ ≤ 2, we have We choose β =2p/ðp + 1Þ < 2 to get Combining (20) and (23), the result was obtained.

Lemma 3.
There exists a positive constant C > 0 depending on ½0, L only, such that for any u ∈ H 1 0 ð0, LÞ, provided that Proof. We set By using the inequalitieskuk 2 2 ≤ Ckuk 2 p ≤ Cðkuk p p Þ 2/p , we have the following corollary.
Proof. If kuk p ≥ 1, then If kuk p ≤ 1, then kuk s p ≤ kuk 2 p : Using the Sobolev embedding theorems, we have Now we are ready to state and prove our main result. For 3 Advances in Mathematical Physics this purpose, we define Corollary 6. Assume that (18) holds. Then for any u ∈ ðH 1 0 ð0, LÞÞ n and 2 ≤ s ≤ p: Theorem 7. Assume that (18) holds. Assume further that Then, the solution of (12) blows up in finite time.

Conclusion
In this work, we are interested 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. 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. In the future work, we will try to study the local existence for this problem with respect to some proposal conditions.