非线性发展方程解的衰减性质研究
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摘要
本文研究了以下三个方程解的衰减性：以及utt-div(q(x)(?)u)+u|u|p-2=0.其中u是关于变量为t,x的函数,即u=u(t,x);g(.)表示记忆项以指数衰减的正核函数.
本文首先证明了第一个方程的解在a=0以及a≠0两种情况下均呈指数衰减性；然后通过构造合适的Lyaponov函数讨论得出第二个方程的解呈指数衰减性；最后则是运用Matsuyama[19]和Todorova-Yordanov[20]的处理方法研究得出第三个方程的解同样具备衰减性.
In this paper we obtain an exponential rate of decay for the solution of equations by constructing a suitable Lyapunov function: And a local energy result is derived to the following nonlinear wave equation using the technique of Matsuyama[19]and Todorova-Yordanov[20]: utt-div(k(x)(?)u)+u|u|p-2=0. Here the kernel g in the memory term decays exponentially and u is a function about the variable of t,x.
本文首先证明了第一个方程的解在a=0以及a≠0两种情况下均呈指数衰减性；然后通过构造合适的Lyaponov函数讨论得出第二个方程的解呈指数衰减性；最后则是运用Matsuyama[19]和Todorova-Yordanov[20]的处理方法研究得出第三个方程的解同样具备衰减性.
In this paper we obtain an exponential rate of decay for the solution of equations by constructing a suitable Lyapunov function: And a local energy result is derived to the following nonlinear wave equation using the technique of Matsuyama[19]and Todorova-Yordanov[20]: utt-div(k(x)(?)u)+u|u|p-2=0. Here the kernel g in the memory term decays exponentially and u is a function about the variable of t,x.
引文
[1]Jishan Fan, Hongwei Wu, Exponential Decay For The Scmilinear Wave Equation With Source Teams[J]. Electronic Journal of Differential Equa-tions,2006,82:1-6.
[2]E.Zuazua, Exponential Decay For The Semilinear Wave Equation With Locally Distributed Damping[J]. Communications in Partial Differential Equations,1990,15:205-235.
[3]韩小森,工明新,带非线性阻尼的粘弹方程解得整体存在性和一致衰减性[J].数学年刊,2009,30：31-42.
[4]Ryo Ikehata, Local energy decay for linear wave equations with variable coefficients[J]. Mathematical Analysis And Applications,2005,306:330-348.
[5]C.Morawetz, The decay of solutions of the exerior initial-boundary value problem for the wave equation[J]. Pure and Applied Mathematics,1961, 14:561-568.
[6]Ryo Ikehata, Local energy decay for linear wave equations with non-compactly supported initial data[J]. Mathematical Methods in the Applied Sciences,2004,27:1881-1892.
[7]Marcelo M.Cavalcant, Exponential Decay For The Solution Of Semilinear Viscoelastic Wave Equation With Localized Damping[J]. Electronic Journal Of Differential Equations.2002.2002:1-14.
[8]Qiaozhen Ma, Chengkui Zhong, Existence Of Strong Global Attractors For Hyperbolic Equation With Linear Memory[J]. Applied Mathematics and Computation.,2004,157:745-758.
[9]Stephane Gerbi.Beckacem Said-Hohari, Exponential Decay For Solutions To Semilinear Damped Wave Equation[J]. a paraitre dans screte and Continuous Dynamical Systems,(2011).[HAL][arxiv].
[10]A.Haraux,E.Zuazua. Decay estimates for some semilinear damped hyper-bolic problems[J]. Arch. Rational Merck. Anal,1988,150:191-206.
[11]J.E.Munoz Rivera, Global solution on a quasilinear wave equation with memory [J]. Bolletino dell Unione Mathematica Italiana.1994,7:289-303.
[12]Ryo Ikehata, Two Dimensional Exterior Mixed Problem For Semilinear Damped Wave Equations[J]. Mathematical Analysis And Applications,2005, 301:366-377.
[13]F.Gazzola,M-Sqnassina, Global solutions and finite time blow up for damped semilinear wave equation[J]. Ann.I.H.Poincare,2006,23:185-207.
[14]Mitsuhiro Nakao, Decay Of Solutions Of The Wave Equation With Some Localized Dissipations[J]. Nonlinear Analysis,1997,:3775-3784.
[15]R. Quint anilla. Exponential Decay In Mixtures With Localized Dissipative Term[J]. Applied Mathematics Letters,2005,18:1381-1388.
[16]F.Linares, A.F.Pazoto, On The Exponential Decay Of The Critical Gener-alized Korteweg-De vries Equation With Localized Damping[J]. Proceedings Of The American Mathematical Society,2007,135:1515-1522.
[17]江璇,任利宁,带衰退记忆的抽象发展方程全局吸引子的存在性[J].兰州大学学报：自然科学版,2008，44：99-102.
[18]Roger Teman, Infinite-Dimensional Dynamical Systems In Mechanics And Physics[M]. Springer,New York,1997.
[19]Jorge A.Esquivel-Avila, The dynamics of a nonlinear wave equation[J]. Mathematical Analysis And Applications,2003,279:135-150.
[20]Filippo Gazzola. Marco Squassina, Global Solution And Finite Time Blow Up For Damped Semilinear Wave Equation[J]. Ann.I.H.Poincare,2006, 23:185-207.
[21]Ryo Ikehata, Matsuyama, Behaviour of solutions to the heat and wave equations in exterior domains[J]. Sci.math. Japan,2002,55:33-42.
[22]Todorova.Yordanov. Critical exponent for a nonlinear wave equation with damping[J]. J.Differential Equation,2001,174:464-489.
[23]王艳华,蒲志林, Lyaponov函数与方程的指数衰减性[J].四川师范大学:自然科学版,已录用.
[24]秦雨萍,蒲志林,王艳华,Kirchhoff方程的解的衰减性质[J].四川师范大学:自然科学版,已录用.
[25]Zhilin Pu.Yanhua Wang, Local energy decay for nonlinear wave equation. submitted.
[26]Coleman B D,Noll W, Foundations of linear viscoelasticity[J]. Rev Mod Phys,1961,33:239-249.
[27]韩英豪,张俊丽,齐宝新,非线性波动方程的解的存在性和衰减性[J].辽宁师范大学:自然科学版,,2009,(32)：396-400.
[28]方道元,徐江,强阻尼波方程解的整体存在性和一致衰减性[J].数学物理学报,2006,(26)：753-765.
[29]杨芳,高洪俊,带有不定阻尼的一维非线性波动方程的指数衰减性[J].南京师大学报:自然科学版,2009,(32)：17-21.
[30]钟承奎,牛明飞,关于无穷维耗散非线性动力系统全局吸引子存在性[J].兰州大学学报:自然科学版,2003,(39)：1-5.
[31]周蜀林,偏微分方程[M].北京大学出版社,北京,2005.
[32]廖为,蒲志林,一类缺乏紧性的P-Laplacian方程非平凡弱解的存在性[J].四川师范大学：自然科学学报：2006,(29)：26-29.
[33]Dafermos C M, Asymptotic stability in viscoelasticity[J]. Arch Rational Mech,Anal,1970,(37):297-308.
[34]Claudio Giorgi,Vittorino Pata, Global attractors for a semilinear hyper-bolic equation in viscoelasticity. Journal of Mathematical Analysis and Applications,2001, (260):83-99.
[2]E.Zuazua, Exponential Decay For The Semilinear Wave Equation With Locally Distributed Damping[J]. Communications in Partial Differential Equations,1990,15:205-235.
[3]韩小森,工明新,带非线性阻尼的粘弹方程解得整体存在性和一致衰减性[J].数学年刊,2009,30：31-42.
[4]Ryo Ikehata, Local energy decay for linear wave equations with variable coefficients[J]. Mathematical Analysis And Applications,2005,306:330-348.
[5]C.Morawetz, The decay of solutions of the exerior initial-boundary value problem for the wave equation[J]. Pure and Applied Mathematics,1961, 14:561-568.
[6]Ryo Ikehata, Local energy decay for linear wave equations with non-compactly supported initial data[J]. Mathematical Methods in the Applied Sciences,2004,27:1881-1892.
[7]Marcelo M.Cavalcant, Exponential Decay For The Solution Of Semilinear Viscoelastic Wave Equation With Localized Damping[J]. Electronic Journal Of Differential Equations.2002.2002:1-14.
[8]Qiaozhen Ma, Chengkui Zhong, Existence Of Strong Global Attractors For Hyperbolic Equation With Linear Memory[J]. Applied Mathematics and Computation.,2004,157:745-758.
[9]Stephane Gerbi.Beckacem Said-Hohari, Exponential Decay For Solutions To Semilinear Damped Wave Equation[J]. a paraitre dans screte and Continuous Dynamical Systems,(2011).[HAL][arxiv].
[10]A.Haraux,E.Zuazua. Decay estimates for some semilinear damped hyper-bolic problems[J]. Arch. Rational Merck. Anal,1988,150:191-206.
[11]J.E.Munoz Rivera, Global solution on a quasilinear wave equation with memory [J]. Bolletino dell Unione Mathematica Italiana.1994,7:289-303.
[12]Ryo Ikehata, Two Dimensional Exterior Mixed Problem For Semilinear Damped Wave Equations[J]. Mathematical Analysis And Applications,2005, 301:366-377.
[13]F.Gazzola,M-Sqnassina, Global solutions and finite time blow up for damped semilinear wave equation[J]. Ann.I.H.Poincare,2006,23:185-207.
[14]Mitsuhiro Nakao, Decay Of Solutions Of The Wave Equation With Some Localized Dissipations[J]. Nonlinear Analysis,1997,:3775-3784.
[15]R. Quint anilla. Exponential Decay In Mixtures With Localized Dissipative Term[J]. Applied Mathematics Letters,2005,18:1381-1388.
[16]F.Linares, A.F.Pazoto, On The Exponential Decay Of The Critical Gener-alized Korteweg-De vries Equation With Localized Damping[J]. Proceedings Of The American Mathematical Society,2007,135:1515-1522.
[17]江璇,任利宁,带衰退记忆的抽象发展方程全局吸引子的存在性[J].兰州大学学报：自然科学版,2008，44：99-102.
[18]Roger Teman, Infinite-Dimensional Dynamical Systems In Mechanics And Physics[M]. Springer,New York,1997.
[19]Jorge A.Esquivel-Avila, The dynamics of a nonlinear wave equation[J]. Mathematical Analysis And Applications,2003,279:135-150.
[20]Filippo Gazzola. Marco Squassina, Global Solution And Finite Time Blow Up For Damped Semilinear Wave Equation[J]. Ann.I.H.Poincare,2006, 23:185-207.
[21]Ryo Ikehata, Matsuyama, Behaviour of solutions to the heat and wave equations in exterior domains[J]. Sci.math. Japan,2002,55:33-42.
[22]Todorova.Yordanov. Critical exponent for a nonlinear wave equation with damping[J]. J.Differential Equation,2001,174:464-489.
[23]王艳华,蒲志林, Lyaponov函数与方程的指数衰减性[J].四川师范大学:自然科学版,已录用.
[24]秦雨萍,蒲志林,王艳华,Kirchhoff方程的解的衰减性质[J].四川师范大学:自然科学版,已录用.
[25]Zhilin Pu.Yanhua Wang, Local energy decay for nonlinear wave equation. submitted.
[26]Coleman B D,Noll W, Foundations of linear viscoelasticity[J]. Rev Mod Phys,1961,33:239-249.
[27]韩英豪,张俊丽,齐宝新,非线性波动方程的解的存在性和衰减性[J].辽宁师范大学:自然科学版,,2009,(32)：396-400.
[28]方道元,徐江,强阻尼波方程解的整体存在性和一致衰减性[J].数学物理学报,2006,(26)：753-765.
[29]杨芳,高洪俊,带有不定阻尼的一维非线性波动方程的指数衰减性[J].南京师大学报:自然科学版,2009,(32)：17-21.
[30]钟承奎,牛明飞,关于无穷维耗散非线性动力系统全局吸引子存在性[J].兰州大学学报:自然科学版,2003,(39)：1-5.
[31]周蜀林,偏微分方程[M].北京大学出版社,北京,2005.
[32]廖为,蒲志林,一类缺乏紧性的P-Laplacian方程非平凡弱解的存在性[J].四川师范大学：自然科学学报：2006,(29)：26-29.
[33]Dafermos C M, Asymptotic stability in viscoelasticity[J]. Arch Rational Mech,Anal,1970,(37):297-308.
[34]Claudio Giorgi,Vittorino Pata, Global attractors for a semilinear hyper-bolic equation in viscoelasticity. Journal of Mathematical Analysis and Applications,2001, (260):83-99.