Existence and non-existence of global solutions for the wave equations

doi:10.3969/j.issn.1000-5641.2018.02.001

Abstract

Abstract: In this paper we investigated the initial boundary value problem for a class of higher order wave equations with two opposite source terms. Firstly, we introduced the latest research progress of the wave equations and defined some important generalized functionals and sets, then the properties of the functionals were discussed. Secondly, it was proved that these sets were invariant under the wave equation. Finally, we proved the existence of global weak solutions by the combination of Galerkin approximation method and potential well method, and obtained the conditions of the non-existence of global weak solutions by using the potential well method and the convexity. The optimal threshold results were given for the existence and non-existence of global weak solutions.

Key words: wave equation, k-Laplace operator, existence, non-existence, potential well

CLC Number:

O175.29

JIN Shou-bo, ZHANG Zu-feng. Existence and non-existence of global solutions for the wave equations[J]. Journal of East China Normal University(Natural Sc, 2018, 2018(2): 1-10.

References

[1] SATTINGER D H. On global solution of nonlinear hyperbolic equations[J]. Archivef or Rational Mechanics and Analysis, 1968, 30:148-172.
[2] PAYNE L E, SATTINGER D H. Sadle points and instability of nonlinear wave equations[J]. Israel Journal of Mathematics, 1975, 22:273-303.
[3] LIU Y C. On potential wells and vacuum is olating of solutions for semilinear wave equations[J]. Journal of Differential Equations, 2003, 192(1):155-169.
[4] LIU Y C, LI P. On potential well and application to strong damped nonlinear wave equations[J]. Acta Math Appl Sin, 2004, 27(4):523-536.
[5] 徐润章, 沈继红, 刘亚成. 位势井及其对具有异号源项波动方程的应用[J]. 工程数学学报, 2007, 24(5):931-934.
[6] XU R Z, YU T. Remarks on wave equations involving two opposite nonlinear source terms[J]. J Appl Math Comput, 2009, 29(29):15-18.
[7] 叶朝辉, 罗显康. 具有两个异号非线性源项的波动方程的整体解[J]. 西南民族大学学报(自然科学版), 2007, 33(4):718-721.
[8] MESSAOUDI S A, SAID H. Global nonexistence of positve initial energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms[J]. J Math Anal Appl, 2010, 365:277-287.
[9] RAMMAHA M A, SAKUNTASATHIEN S. Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms[J]. Nonlinear Analysis, 2010, 72:2658-2683.
[10] SHANG Y D. Initial boundary value problem of equation u_tt -△u -△u_t -△u_tt=f(u)[J]. Acta Math Appl Sin, 2000, 23:385-393.
[11] CHUESHOR L. Long-time dynamics of Kirchhoff wave models with strong nonlinear damping[J]. Journal of Differential Equations, 2012, 252:1229-1262.
[12] 狄华斐, 尚亚东. 一类带有非线性阻尼项和源项的四阶波动方程整体解的存在性与不存在性[J]. 数学物理学报, 2015, 35A(3):618-633.
[13] 黄文毅, 张健. 具有强阻尼项和非线性阻尼项的波动方程解的整体存在性和有限时间爆破[J]. 应用数学, 2008, 21(4):787-793.
[14] LIN Q, WU Y H, LAI S Y. On global solution of an initial boundary value problem for a class of damped nonlinear equations[J]. Nonlinear Analysis, 2008, 69:4340-4351.
[15] 徐润章, 刘博为. 四阶具强阻尼非线性波动方程解的整体存在性与不存在性[J]. 数学年刊, 2011, 32A(3):267-276.
[16] VESA J, PETRI J. A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation[J]. Communications in Partial Differential Equation, 2012, 37(5):934-946.
[17] GAO D M, PENG S J, YAN S S. Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth[J]. Journal of Functional Analysis, 2012, 262(6):2861-2902.
[18] DI H F, SHANG Y D, PENG X M. Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term[J]. Mathematische Nachrichten, 2016, 289(3):1408-1432.
[19] DI H F, SHANG Y D. Cauchy problem for a higher order generalized Boussinesq-type equation with hydrodynamical damped term[J]. Applicable Analysis:An International Journal, 2016, 95(3):690-714.
[20] LIN Y C, XU R Z. Forth order wave equations with nonlinear strain and source terms[J]. J Math Anal Appl, 2007, 331:585-607.
[21] LIONS J L. Quelques méthodes de résolution des problémes aux limites non linéaires[M]. Paris:Dounod Gauthier-Villars, 1969:4-27.