Research Article Blow-Up of Certain Solutions to Nonlinear Wave Equations in the Kirchhoff-Type Equation with Variable Exponents and Positive Initial Energy Loay Alkhalifa , 1 Hanni Dridi, 2 and Khaled Zennir 1,3 1 Department of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia 2 Laboratory of Applied Mathematics, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria 3 Laboratoire de Mathématiques Appliquées et de Modélisation, Université 8 Mai 1945 Guelma, B.P. 401 Guelma 24000, Algeria Correspondence should be addressed to Loay Alkhalifa; loay.alkhalifa@qu.edu.sa Received 2 March 2021; Revised 16 March 2021; Accepted 22 March 2021; Published 8 April 2021 Academic Editor: Liliana Guran Copyright © 2021 Loay Alkhalifa et al. 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. This paper is concerned with the blow-up of certain solutions with positive initial energy to the following quasilinear wave equation: u tt − MðN uðt ÞÞΔ pð·Þ u + gðu t Þ = f ðuÞ. This work generalizes the blow-up result of solutions with negative initial energy. 1. Introduction Let Ω be an open bounded Lipschitz domain in ℝ n ðn ≥ 1Þ, T > 0, Q T = Ω × ð0, T Þ. We consider the following nonlinear hyperbolic equation: u tt − MN ut ðÞ ð ÞΔ px ðÞ u + gu t ð Þ = fu ðÞ, x, t ð Þ ∈ Q T , ux, t ð Þ = 0, x ∈ ∂Ω, t ∈ 0, T ð Þ, ux,0 ð Þ = u 0 x ðÞ, u t x,0 ð Þ = u 1 x ðÞ, x ∈ Ω: 8 > > < > > : ð1Þ Here, ∂Ω is a Lipschitz continuous boundary. The initial conditions meet the following: u 0 ∈ W 1,p · ðÞ 0 Ω ð Þ, u 1 ∈ L 2 Ω ð Þ: ð2Þ The Kirchhoff function M : ℝ + ⟶ ℝ + is continuous and has the standard form: M τ ðÞ = a + bγτ γ−1 , a, b ≥ 0, γ ≥ 1, a + b > 0, γ > 1 if b >0: ð3Þ The elliptic nonhomogeneous pðxÞ-Laplacian operator is defined by Δ px ðÞ u = ∇ · ∇u j j px ðÞ−2 ∇u   , ð4Þ where ∇· is the vectorial divergence and ∇ is the gradient of u. The functional N ut ðÞ = ð Ω ∇u j j px ðÞ px ðÞ dx, ð5Þ is the naturally associated pðxÞ-Dirichlet energy integral. The term with a variable exponent fu ðÞ = cx, t ð Þ u jj qx ðÞ−2 u, ð6Þ plays the role of a source, and the dissipative term with a variable exponent gu t ð Þ = dx, t ð Þ u t j j rx ðÞ−2 u t , ð7Þ is a strong damping term. Hindawi Journal of Function Spaces Volume 2021, Article ID 5592918, 9 pages https://doi.org/10.1155/2021/5592918