一类具对数源项波动方程的初边值问题

Abstract

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

CLC Number:

O175.4

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.

Trendmd

References

[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

Initial Boundary Value Problem for a Class Wave Equation with Logarithmic Source Term

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Liu Gongwei, Diao Lin. On Convexity for Energy Decay Rates of the Second-Order Evolution Equation with Memory and Time-Varying Delay [J]. Acta mathematica scientia,Series A, 2018, 38(3): 527-542.
[2]	Ye Yaojun, Hu Yue. Global Existence and Exponential Decay of Solution for Some Higher-Order Wave Equation [J]. Acta mathematica scientia,Series A, 2018, 38(1): 110-121.
[3]	Feng Baowei, Su Keqin. Uniform Decay for a Fourth-Order Viscoelastic Equation with Density in Rⁿ [J]. Acta mathematica scientia,Series A, 2018, 38(1): 122-133.
[4]	Chen Chih-Wei. A Note on Geodesic Loops in Complete Non-Compact Ricci Solitons [J]. Acta mathematica scientia,Series A, 2017, 37(1): 1-6.
[5]	Pei Ruichang, Zhang Jihui, Ma Caochuan. Nontrivial Solutions for a Class of Superlinear (p, 2)-Laplacian Dirichlet Problems [J]. Acta mathematica scientia,Series A, 2017, 37(1): 92-101.
[6]	Yu Pei. Decay of Weak Solutions to the Multi-Dimensional Burgers' Equation with Fractional Diffusion [J]. Acta mathematica scientia,Series A, 2016, 36(2): 340-352.
[7]	Yang Fen. Decay Rate of Solutions to An Inhomogenerous Semilinear Biharmonic Equation in Entire Space [J]. Acta mathematica scientia,Series A, 2015, 35(2): 282-287.
[8]	ZHANG Chun-Guo, ZHANG Hai-Yan, GU Shang-Wu. Energy Decay Estimates for the Weakly Coupled Systems with Local Memory Damping [J]. Acta mathematica scientia,Series A, 2015, 35(1): 194-209.
[9]	ZHU Xu-Sheng, LI Fang-E. The Global Solutions of the Compressible Euler Equations With Nonlinear Damping in 3-Dimensions [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1111-1122.
[10]	LIU Shuai, CHEN Jian-He, ZHOU Zhe-Yan. Asymptotic Solution for Second-order Semilinear Singularly Perturbed Boundary Value Problems [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1104-1110.
[11]	ZHANG Dan-Dan. Pointwise Estimates for the Solution to the Multi-Dimensional Generalized BBM-Burgers Equation [J]. Acta mathematica scientia,Series A, 2014, 34(3): 473-486.
[12]	Jum-Ran KANG. GENERAL DECAY FOR A DIFFERENTIAL INCLUSION OF KIRCHHOFF TYPE WITH A MEMORY CONDITION AT THE BOUNDARY [J]. Acta mathematica scientia,Series A, 2014, 34(3): 729-738.
[13]	LIU Shuai, ZHOU Zhe-Yan, SHEN Jian-He. Asymptotic Behavior of Solutions for Second-Order Semilinear Singularly Perturbed Boundary Value Problem [J]. Acta mathematica scientia,Series A, 2014, 34(2): 437-444.
[14]	SHI Qi-Hong. Uniqueness of the Solution and Energy Decay for Nonlinear KGS System [J]. Acta mathematica scientia,Series A, 2013, 33(5): 842-849.
[15]	LI Zhong-Ping, XU Si, DU Wan-Juan. Secondary Critical Exponent and Life Span for a Fast Diffusion Equation [J]. Acta mathematica scientia,Series A, 2012, 32(5): 904-913.