A global nonexistence of solutions for a quasilinear viscoelastic wave equation with acoustic boundary conditions

Abstract

In this paper, we consider a quasilinear viscoelastic wave equation with acoustic boundary conditions. Under some appropriate assumption on the relaxation function g, the function , p > max{ρ + 2, m, q, 2}, and the initial data, we prove a global nonexistence of solutions for a quasilinear viscoelastic wave equation with positive initial energy.

Global solutions of wave equations with multiple nonlinear source terms under acoustic boundary conditions

“…Remark 3 Compared with the quasilinear viscoelastic wave equation in [8] and [9], we not only prove the existence of global solutions of the equation, but also give the more complex nonlinear terms.…”

“…Many results have been obtained on the existence of global solutions for wave equations with acoustic boundary conditions [3,5,7,8,12], the Faedo-Galerkin method is one of the most used methods to prove the existence of solutions. [25] and [26] respectively proved the existence of solutions for two different kinds of wave equations with acoustic boundary conditions by using the Faedo-Galerkin method, but these two kinds of equations contain neither multiple nonlinear source terms nor m and β order Laplacian operators.…”

“…The existence and nonexistence of global solutions for viscoelastic wave equations are analyzed in [6]. Nonlinear and quasi-linear viscoelastic wave equations are studied in [7] and [8], respectively. Jeong et al obtained the nonexistence conditions of global solutions for a class of quasi-linear wave equations with acoustic boundary conditions by using the energy functional method in [9].…”

Exponential stabilization of wave equation with acoustic boundary conditions and its application to memory‐type boundary

“…Recently, wave equations with acoustic boundary conditions have been studied by many authors. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] Compared with previous articles on this subject, the main difference of this article is related with Equations (1) 3 and (2) 3 . In most previous work, for instance, 9,[12][13][14]16,17,19 the memory term of 𝜕 𝜈 u t has not been considered.…”

“…The asymptotic behavior was obtained in Beale 3 (Theorem 2.6). Recently, wave equations with acoustic boundary conditions have been studied by many authors 5–19 …”