一类具对数源项波动方程的初边值问题

摘要/Abstract

摘要：

该文考虑一类具对数源项波动方程的初边值问题.利用Galerkin方法结合对数Sobolev不等式和对数Gronwall不等式，对所有初始值得到了整体解的存在性.通过引入位势井，给出了解在时间无穷远处爆破（即指数增长）的充分条件.当具有对数源项的波动方程还带有线性阻尼时，通过构造适当的Lyapunov函数，得到了能量的衰减估计.

关键词: 对数波动方程, 整体存在性, 指数增长, 衰减

Abstract:

In this paper we consider the initial boundary value problem for a class wave equation with logarithmic source term. By using Galerkin method combining with the logarithmic Sobolev inequality and logarithmic Gronwall inequality, we obtain the existence of global solution for all initial data. By introducing potential well theory, we give the sufficient condition of the blow-up property in infinity time (i.e. exponential growth) of the solution. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for the wave equation with logarithmic source term and linear damping term.

Key words: Logarithmic wave equation, Existence of global solution, Exponential growth, Decay

中图分类号:

O175.4

张宏伟, 刘功伟, 呼青英. 一类具对数源项波动方程的初边值问题[J]. 数学物理学报, 2016, 36(6): 1124-1136.

Zhang Hongwei, Liu Gongwei, Hu Qingying. Initial Boundary Value Problem for a Class Wave Equation with Logarithmic Source Term[J]. Acta mathematica scientia,Series A, 2016, 36(6): 1124-1136.

参考文献

[1] Bloom F. Asymptotic behavior of solutions to the damped nonlinear equation (∂²)/(∂t²)u(x,t)+r(∂)/(∂t)u(x,t)-(∂)/(∂x)σ((∂)/(∂x)u(x,t))=0. J Math Anal Appl, 1982, 87:551-559
[2] Yang Z J. Existence and nonexistence of global solutions to a generalized modification of the improved Boussinesq equation. Math Meth Appl Sci, 1988, 21:1467-1477
[3] Wazwaz A M. Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation. Ocean Engineering, 2015, 94:111-115
[4] Liu Y. Decay and scattering of small solutions of a generalized Boussinesq equation. J Funct Anal, 1997, 147:51-68
[5] Lin Q, Wu Y H, Loxton R. On the Cauchy problem for a generalized Boussinesq equation. J Math Anal Appl, 2009, 353:186-195
[6] Bona J, Sachs R. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Comm Math Phys, 1988, 118:15-29
[7] Yang Z J. Existence and nonexistence of global solutions to a generalized modification of the improved Boussinesq equation. Math Meth Appl Sci, 1998, 21:1467-1477
[8] Chen G W, Zhang H W. Initial boundary value problem for a system of generalized IMBq equation. Math Meth Appl Sci, 2004, 27:497-518
[9] Chen G W, Wang Y P, Wang S B. Initial boundary value problem of the generalized cubic double dispersion equation. J Math Anal Appl, 2004, 299:563-577
[10] You Y C. Nonlinear exponential stabilization of damped Boussinesq equation//Bensoussa A, Lions J L. Analysis and Optimization of Systems. New York:Springer, 1990:642-651
[11] Varlamov V V. On the initial-boundary value for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 1998, 4(3):431-444
[12] Lai S Y, Wu Y H, Xu Y. The global solution of an initial boundary value problem for the damped Boussinesq equation. Commun Pure Appl Anal, 2004, 3(2):319-328
[13] Xue R Y. The initial boundary value problem for the goog Boussinesq equation on the bounded domain. J Math Anal Appl, 2008, 343:975-995
[14] Liu Y C, Xu R Z. Potential well method for initial boundary value problem of the generalized double dispersion equations. Commun Pure Appl Anal, 2008, 7:63-81
[15] Li K, Fu S H. Asymptotic behavior for the damped Boussinesq equation with critical nonlinearity. Applied Mathematics Letters, 2014, 30:44-50
[16] Yang Z J. Longtime dynamics of the damped Boussinesq equation. J Math Anal Appl, 2013, 399:180-190
[17] Zhuang W, Yang G T. Propagation of solitary waves in the nonlinear rods. Applied Mathematics and Mechanics, 1986,7:571-581
[18] Zhang S Y, Zhuang W. Strain solitary waves in the nonlinear elastic rods. Acta Mechanica sinica, 1988, 20:58-66
[19] Chen G W, Lu B. The initial-boundary value problems for a class of nonlinear wave equations with damping term. J Math Anal Appl, 2009, 351:1-15
[20] Xu R Z, Wang S, Yang Y B, Din Y H. Initial boundary value problem for a class of fourth-order wave equation with viscous damping term. Applicable Analysis, 2013, 92(7):1403-1416
[21] 张宏伟, 陈国旺. 一类非线性四阶波动方程的位势井方法. 数学物理学报, 2003, 23A(6):758-768Zhang H W, Chen G W. Potential well method for a class of nonlinear wave equations of fourth-order. Acta Math Sci, 2003, 23A(6):758-768
[22] 杨志坚, 陈国旺. 具有阻尼项的非线性波动方程的初值问题. 应用数学学报, 2000, 23:45-54Yang Z J, Chen G W. Initial value problem for a nonlinear wave equation with damping term. Acta Math Appl Sci, 2000, 23:45-54
[23] Chen G, Yang Z J. Existence and non-existence of global solutions for a class of nonlinear wave equations. Math Meth Appl Sci, 2000, 23:615-631
[24] Yang Z J. Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term. J Differ Equas, 2003, 187:520-540
[25] Liu Y C, Xu R Z. Fourth order wave equations with nonlinear strain and source terms. J Math Anal Appl, 2007, 331:585-607
[26] Liu Y C, Xu R Z. A class of fourth order wave equations with dissipative and nonlinear strain terms. J Differ Equas, 2008, 244:200-228
[27] Yang Z J. Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow. J Math Anal Appl, 2006, 313:197-217
[28] Zhang H W, Hu Q Y. Global existence and nonexistence of solution for Cauchy problem of two-dimensional generalized Boussinesq equation. J Math Anal Appl, 2015, 422:1116-1130
[29] Kutev N, Kolkovska N, Dimova M. Global existence of Cauchy problem for Boussinesq equation with combined power-type nonlinearities. J Math Anal Appl, 2014, 410:427-444
[30] Xu R Z. Cauchy problem for generalized Boussinesq equation with combined power-type nonlinearities. Math Meth Appl Sci, 2011, 34:2318-2328
[31] Chen H, Luo P, Liu G W. Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity. J Math Anal Appl, 2015, 422:84-98
[32] Chen H, Tian S Y. Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J Differ Equas, 2015, 258:4424-4442
[33] Cazenave T, Haraux A. Equations devolution avec non-linearite logarithmique. Ann Fac Sci Toulouse Math, 1980, 2(1):21-51
[34] Gorka P. Logarithmic Klein-Gordon equation. Acta Phys Polon B, 2009, 40(1):59-66
[35] Lieb E, Loss M. Analysis (second ed). Providence:Amer Math Soc, 2001
[36] Cotsiolis A, Tavoularis N K. On logarithmic Sobolev inequalities for higher order fractional derivatives. C R Acad Sci Paris, 2005, 340:205-208
[37] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math, 1975, 22:273-303
[38] Liu Y C. On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Analysis, 2006, 64:2665-2687
[39] Haraux A, Zuazua E. Decay estimates for some semilinear damped hyperbolic problems. Arch Ration Mech Anal, 1988, 150:191-206
[40] Zuazua E. Exponential decay for the semilinear wave equation with locally distributed damping. Comm Partial Differ Equas, 1990, 15(2):205-235
[41] Benaissa A, Messaoudi S. Exponential decay of solutions of a nonlinearly damped wave equation. Nonlinear Differential Equations Applications, 2005, 12(4):391-399
[42] Gerbi S, Said-Houari B. Exponential decay for solutions to semilinear damped wave equation. Discrete and Continuous Dynamical Systems, 2012, 5(3):559-566