Global Nonexistence of Solutions for Viscoelastic Wave Equations of Kirchhoff Type with High Energy

Abstract

In this paper we consider the viscoelastic wave equation of Kirchhoff type:with Dirichlet boundary conditions. Under some suitable assumptions on g and the initial data, we established a global nonexistence result for certain solutions with arbitrarily high energy.

Nonexistence of global solutions for a class of viscoelastic wave equations

“…In [31] the blow-up property the problem (KE) with nonlinear damping is analyzed for E 0 < d and I(u 0 ) < 0. For high initial energies the blow-up of solutions is proved in [12] and [7] under the conditions given in (17), and the same property is showed in [34] if (19) is satisfied. Consequently, the same remarks made in the previous examples regarding these conditions and our contribution apply to this problem.…”

“…Applications. For concrete problems of the type (P) some authors have studied the blow-up for arbitrary positive values of the initial energy, see [7,8,12,13,15,19,20], [27]- [34]. We next present and comment the application of our main result to some concrete problems, such as the wave, Kirchhoff, and Petrovsky equations with memory.…”

“…This problem has been studied in [7,12,31,34] within the framework of strong solutions. That is, the initial data are in the following space (u 0 , u 1 ) ∈ (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) ⊂ H ≡ H 1 0 (Ω) × L 2 (Ω).…”

“…Generalizations of the concavity method have been utilized to obtain sufficient conditions to get blow-up for arbitrary positive values of the initial energy. Two notable first articles on this subject are [30,29], see also [7,8,12,13,15,19,20,27,28], [31]- [34]. Related and interesting results have been reported in [2]- [5], [14], [16]- [18], [21,24,26,35].…”

Blow-up of arbitrarily positive initial energy solutions for a viscoelastic wave system with nonlinear damping and source terms

“…Besides, for the work on quasilinear wave equations, we refer the reader to [16][17][18] and the references therein.…”