Existence and nonexistence of global solutions for nonlinear transmission acoustic problem

Existence and nonexistence of global solutions for nonlinear transmission acoustic problem

In this paper we consider a mixed problem for the nonlinear wave equations with transmission acousticconditions, that is, the wave propagation over bodies consisting of two physically different types of materials, one ofwhich is clamped. We prove the existence of a global solution. Under the condition of positive initial energy we showthat the solution for this problem blows up in finite time.

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  • [1] Aliev AB, Mammadhasanov EH. Well-posedness of initial boundary value problem on longitudinal impact on a composite linear viscoelastic bar. Math Method Appl Sci 2017; 40: 5380-5390.
  • [2] Bae JJ. Nonlinear transmission problem for wave equation with boundary condition of memory type. Acta Appl Math 2010; 110: 907-919.
  • [3] Ball J. Remarks on blow up and nonexistence theorems for nonlinear evolution equations. Quart J Math Oxford 1977; 28: 473-486.
  • [4] Bastos WD, Raposo CA. Transmission problem for waves with frictional damping. Electron J Differ Eq 2007; 60: 1-10.
  • [5] Beale JT. Spectral properties of an acoustic boundary condition. Indiana Univ Math J 1976; 25: 895-917.
  • [6] Beale JT. Acoustic scattering from locally reacting surfaces. Indiana Univ Math J 1977; 26: 199-222.
  • [7] Beale JT, Rosencrans SI. Acoustic boundary conditions. B Am Math Soc 1974; 80: 1276-1278.
  • [8] Bilgin BA, Kalantarov VK. Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations. J Math Anal Appl 2013; 403: 89-94.
  • [9] Boukhatem Y, Benabderrahmane B. Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions. Nonlinear Anal 2014; 97: 191-209.
  • [10] Boukhatem Y, Benabderrahmane B. Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions. Acta Math Sin 2016; 32: 153-174.
  • [11] Bucci F, Lasiecka I. Exponential decay rates for structural acoustic model with an over damping on the interface and boundary layer dissipation. Appl Anal 2002; 81: 977-999.
  • [12] Cousin AT, Frota CL, Larkin NA. On a system of Klein-Gordon type equations with acoustic boundary conditions. J Math Appl 2004; 293: 293-309.
  • [13] Dautray R, Lions JL. Analyse et Calcul Numerique pour les Sciences et les Techniques. Paris, France: Masson, 1984 (in French).
  • [14] Frota CL, Cousin AT, Larkin NA. Global solvability and asymptotic behaviour of a hyperbolic problem with acoustic boundary conditions. Funkcial Ekvac 2001; 44: 471-485.
  • [15] Frota CL, Goldstein JA. Some wave equations with acoustic boundary conditions. J Differ Equations 2000; 164: 92-109.
  • [16] Frota CI, Larkin NA. Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains. Prog Nonlin 2005; 66: 297-312.
  • [17] Frota CI, Medeyros LA, Vicente A. A mixed problem for semilinear wave equations with acoustic boundary conditions in domains with non-locally reacting boundary. Electron J Differ Eq 2014; 2014: 1-14.
  • [18] Frota CL, Vicente A. A hyperbolic system of Klein-Gordon type with acoustic boundary conditions. Int J Pure Appl Math. 2008; 47: 185-198.
  • [19] Gal CG, Goldstein GR, Goldstein JA. Oscillatory boundary conditions for acoustic wave equations. J Evol Equ 2003; 3: 623-635.
  • [20] Gao Y, Liang J, Xiao TJ. Observability inequality and decay rate for wave equations with nonlinear boundary conditions. Electron J Differ Eq 2017; 2017: 1-12.
  • [21] Georgiev V, Todorova G. Existence of a global solution of the wave equation with nonlinear damping and source term. J Differ Equations 1994; 109: 295-308.
  • [22] Graber PJ. Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system. Nonlinear Anal 2010; 73: 3058-3068.
  • [23] Graber PJ. Strong stability and uniform decay of silutions to a wave equation with semilinear porous acoustic boundary conditions. Nonlinear Anal 2011; 74: 3137-3148.
  • [24] Graber PJ, Said-Houari B. On the wave equation with semilinear porous acoustic boundary conditions. J Differ Equations 2012; 252: 4898-4941.
  • [25] Haraux A, Zuazua E. Decay estimates for some semilinear damped hyperbolic problems. Arch Ration Mech Anal 1988; 150: 191-206.
  • [26] Isayeva SE. Existence of local solutions for nonlinear wave equations with transmission acoustic conditions. News of Baku University Series of Physico-mathematical Sciences (in press).
  • [27] Jeong JM, Park JY, Kang YH. Global nonexistence of solutions for a quasilinear wave equation with acoustic boundary conditions. Boundary Value Problems 2017; 2017: 1-10.
  • [28] Kalantarov VK, Ladyzhenskaya OA. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types. J Soviet Math 1978; 10: 53-70.
  • [29] Kopackova M. Remarks on bounded solutions of a semilinear dissipative hyperbolic equation. Comment Math Univ Cardin 1989; 30: 713-719.
  • [30] Lasiecka I. Boundary stabilization of a 3-dimensional structural acoustic model. J Math Pure Appl 1999; 78: 203- 232.
  • [31] Levine HA. Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = Au + F (u) . T Am Math Soc 1974; 192: 1-21.
  • [32] Levine HA. Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J Math Anal 1974; 5: 138-146.
  • [33] Levine HA, Park SR. Global existence and global nonexistence of solutions of the Cauchy problem for a non-linearly damped wave equation. J Math Anal Appl 1998; 228: 181-205.
  • [34] Levine HA, Serrin J. Global nonexistence theorems for quasilinear evolution equations with dissipation. Arch Ration Mech Anal 1997; 137: 341-361.
  • [35] Lions JL. Quelques méthodes de rèsolution des problèmes aux limites non linèaires. Paris, France: Dunod Gaulthier- Villars, 1969 (in French).
  • [36] Lions JL, Magenes E. Non-Homogeneous Boundary Value Problems and Applications. I. Berlin, Germany: Springer Verlag, 1972.
  • [37] Liu W, Williams G. The exponential stability of the problem of transmission of the wave equation. Bull Austral Math Soc 1998; 57: 305-327.
  • [38] Morse PM, Ingard KU. Theoretical Acoustics. New York, NY, USA: McGraw-Hill, 1968.
  • [39] Mugnolo D. Abstract wave equations with acoustic boundary conditions. Math Nachr 2006; 279: 299-318.
  • [40] Muñoz Rivera JE, Portillo Oquendo H. The transmission problem of viscoelastic waves. Acta Applicandae Mathematicae 2000; 60: 1-21.
  • [41] Park JY, Ha TG. Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions. J Math Phys 2009; 50: 1-18.
  • [42] Park JY, Park SH. Decay rate estimates for wave equations of memory type with acoustic boundary conditions. Nonlinear Anal 2011; 74: 993-998.
  • [43] Vicente A. Wave equation with acoustic/memory boundary conditions. Bol Soc Parana Mat 2009; 27: 29-39.
  • [44] Vitillaro E. Global existence theorems for a class of evolution equations with dissipation. Arch Ration Mech Anal 1999; 149: 155-182.