Blow-Up of Certain Solutions to Nonlinear Wave Equations in the Kirchhoff-Type Equation with Variable Exponents and Positive Initial Energy

<jats:p>This paper is concerned with the blow-up of certain solutions with positive initial energy to the following quasilinear wave equation: <jats:inline-formula>
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                  </jats:inline-formula>. This work generalizes the blow-up result of solutions with negative initial energy.</jats:p>


Introduction
Let Ω be an open bounded Lipschitz domain in ℝ n ðn ≥ 1Þ, T > 0, Q T = Ω × ð0, TÞ. We consider the following nonlinear hyperbolic equation: Here, ∂Ω is a Lipschitz continuous boundary. The initial conditions meet the following: The Kirchhoff function M : ℝ + ⟶ ℝ + is continuous and has the standard form: The elliptic nonhomogeneous pðxÞ-Laplacian operator is defined by where ∇· is the vectorial divergence and ∇ is the gradient of u. The functional is the naturally associated pðxÞ-Dirichlet energy integral. The term with a variable exponent plays the role of a source, and the dissipative term with a variable exponent is a strong damping term.
The coefficients c and d are continuous in Q T and satisfy where σ is a constant defined in (38). We assume that the Kirchhoff function M, defined by (3), satisfies the following hypotheses: (i) For 1 < α ≤ β < min fn/p + , np − /p + ðn − p − Þg, there exist m 2 ≥ m 1 > 0 such that (ii) For all τ ∈ ℝ + , it holds that The exponents pð·Þ, qð·Þ, and rð·Þ are continuous and satisfy where the constants α and β are given in (10) and Also, we can define p * ðxÞ by +∞, i f p + ≥ n: We also assume that pð·Þ, qð·Þ, and rð·Þ satisfy the log-Hölder continuity condition for L > 0, 0 < δ < 1. Problem (1) models several physical and biological systems such as viscoelastic fluids, filtration processes through a porous medium, and fluids with viscosity dependent on temperature. In the intention of problem (1), we can see that it is linked to the following equation presented by Kirchhoff and Hensel [1] in 1883: The parameters L, h, E, ρ, and P 0 represent, respectively, the length of the string, the area of the cross-section, Young's modulus of the material, the mass density, and the initial tension. This equation is an extension of the classic d'Alembert's wave equation by looking at the effects of changes in the length of the string during the vibrations. As for this problem, it has been studied. More precisely, for gðu t Þ = u t , the global existence and nonexistence results can be found in [2,3], and for gðu t Þ = ju t j p u t , p > 0, the main results of existence and nonexistence are in the paper [4]. In recent years, hyperbolic problems with a constant exponent have been studied by many authors; we refer to interesting works [5][6][7]. However, only a little research has been done regarding hyperbolic problems with nonlinearities of the variable exponent type; some interesting works can be found in [8][9][10][11][12][13].
Recently, in [14], Piskin studied the following wave equation with variable exponent nonlinearities: The author proved, by using the modified energy functional method, the existence of solutions. We have also looked at the asymptotic behavior of the Kirchhoff wave equation problems. We can say that the investigation into the determination of the type, as well as the rate of decay, was the focus of attention of many researchers whose work was represented in [15,16]. Motivated by previous studies, in this work, we consider problem (1), which is more interesting and applicable in the real approach of sciences, so a finite-time blow-up for certain solutions with positive and also negative initial energy has been proved. More precisely, our aim here is to find sufficient conditions on the variable exponents pð·Þ, qð·Þ, and rð·Þ and the initial data for which the blow-up occurs. This paper is organized as follows. After the introduction in the first section, we will give some preliminaries in Section 2. Then, in Section 3, we state the main results which will be proved in Sections 4 and 5.

Preliminaries
Regarding some definitions and basic properties of the generalized Lebesgue-Sobolev spaces L pðxÞ ðΩÞ and W 1,pðxÞ ðΩÞ, where Ω is an open subset of ℝ n , we refer to the book of Musielak [17] and the papers [18,19]. Let For any h ∈ Cð ΩÞ, we write Then, for any pðxÞ ∈ C + ð ΩÞ, we define the variable exponent Lebesgue space as follows: where ϱ pð·Þ is the pð·Þ modular of u, and it is defined by It is equipped with the following so-called Luxemburg norm on this space defined by the formula Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many aspects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if 1 < p − ≤ p + < ∞, and their continuous functions are dense if p + < ∞.

Main Results
Now, we state without proof the following existence result.
Proposition 6. Assume that (2) holds and the coefficients a, b, c, and d satisfy (3) and (9) and the exponents p, q, and r satisfy (12). Then, problem (1) has a unique weak solution such that where p ′ ð·Þ is the conjugate exponent of pð·Þ.
Remark 7. The proof can be established by employing the Galerkin method as in the work of Antontsev [8].
We first define the energy function. Let In order to investigate the properties of EðtÞ, the following lemma is necessary.

Lemma 8.
Suppose that u is a solution of problem (1) that satisfies (29); then, we have Proof. By using the energy function (30) and problem (1), we directly deduce (32).
We also introduce the following lemma.

Lemma 9.
Suppose that the conditions of Lemmas 1-5 hold.
Then, there exists a constant C > 1, which is a generic constant that depends on Ω only, such that for any u ∈ W 1,pð·Þ 0 ðΩÞ and αp − ≤ s ≤ αq − .
Proof. If ϱ qð·Þ ðuÞ > 1, then If ϱ qð·Þ ðuÞ ≤ 1, then we deduce by Lemma 4 that kuk qð·Þ ≤ 1. Then, Lemmas 2 and 5 imply Let B be the best constant of the Sobolev embedding We set Now, the main results of the blow-up for certain solutions with positive/negative initial energy are given by the following theorems.
Theorem 10. Let the assumptions of Proposition 6 be satisfied, and assume that Then, the solutions of (1) blow up in finite time: Theorem 11. Let the assumptions of Proposition 6 be satisfied, and assume that Then, the solution of (1) blows up in finite time (42).

Proof of Theorem 10
To prove Theorem 10, we need the following lemmas.

Lemma 13.
Let the assumptions of Theorem 10 hold. Then, in light of Lemma 12, we have Proof. By using (30), we get Let Lemma 14. Let the assumptions of Theorem 10 be satisfied; then, we have Proof. Using (30), (32), and (51), we obtain Then, the use of (10) gives Now, recalling E 1 in (38), we have On the other hand, we use (9) to get Combining (55) with (56) gives (52).

Proof.
Let
We set for ε small, which will be specified later, and for Now, we are in a position to prove Theorem 10.

Proof of Theorem 11
We set To prove our main result, we first establish the following lemma.
Lemma 18. Let u be the solution of (1). Then, there exists a constant C > 0 such that Proof. Suppose that, by contradiction, there exists a sequence t k such that ∇u t, x ð Þ k k p · ð Þ ⟶ 0, as k ⟶ ∞: Then, by using Lemmas 2 and 5, we get which contradicts the fact that EðtÞ < 0, ∀t ≥ 0.
Using (93) and (94) and applying the same procedures used to prove Theorem 10 will give the proof of Theorem 11.

Data Availability
No data were used in this study.