两类带耗散机制的双曲方程解的适定性研究
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摘要
带耗散机制的非线性双曲方程是一类重要的发展方程,在数学、物理及其它许多领域中都扮演着重要的角色.本文考虑了两类带不同耗散机制的非线性双曲方程,一类是带非线性对流项的阻尼波动方程,另一类是带退化扩散项的守恒律方程.我们研究了这两类方程经典解的整体存在性和大时间行为.本文的主要内容如下：
第一章为绪论.我们首先介绍了课题背景,然后分别回顾了阻尼波动方程和退化双曲抛物方程的物理背景以及研究的历史和现状.最后,我们陈述了本文研究的具体问题及得到的主要结果.
在第二章,我们研究了多维空间中带非线性对流项的阻尼波动方程的Cauchy问题的经典解的整体存在性和Lp(2≤p≤(?))衰减估计.我们考虑方程在常状态u*附近的小扰动问题.首先,利用描述非线性发展方程耗散机制的Shizuta-Kawashima条件,我们发现方程的耗散结构由常状态u*和方程的非线性项共同决定.然后,我们证明了当方程满足某一耗散条件时,其Cauchy问题的整体经典解是存在的.本文采用了李大潜提出的整体迭代法([66]),不需要证明局部解的存在性,而是利用解的衰减性质直接得到了整体经典解的存在性.由于能量估计仅能得到解的有界估计,为得到解的衰减性质,我们使用了高低频分解的办法,将解分为两部分：低频部分和高频部分.对于低频部分,我们使用了Green函数办法.利用线性化方程的Green函数低频部分的估计以及Duhamel原理可以很自然地得到解的低频部分的衰减性质.对于高频部分,我们将能量估计和高频部分有最小频率的性质结合起来,利用Poincare-like不等式和Gronwall不等式也得到了解的衰减性质.最后,我们使用先验估计和频率分解方法得到了解的L2和L(?)的估计,并借助插值不等式得到了解的Lp估计.
在第三章中,我们继续研究多维空间中带非线性对流项的阻尼波动方程.在方程满足耗散条件的前提下,我们运用经典的Green函数方法得到了Cauchy问题解的逐点估计.首先将线性化方程的Green函数分为低频、带宽、高频三部分,再分别估计它们得到Green函数的逐点估计,然后运用Duhamel原理将非线性微分方程变为非线性积分方程,把解表达出来之后再利用先验估计的办法得到非线性问题解的逐点估计.因为非线性项含有解的导数,我们借助了能量估计来封闭高阶导数的估计.逐点估计可以让我们更清晰的看到解的大时间行为,我们发现,波的主体沿着某一条直线移动,并且沿着这条直线衰减最慢.
在第四章,我们试图对非线性退化双曲抛物方程的经典解问题进行研究.我们首先考虑了带退化扩散项的守恒律方程的初边值问题,我们证明了经典解是整体存在的,并且当时间趋于无穷大时解是指数衰减的.我们使用了经典的连续性办法来研究经典解的整体存在性.先证明局部解的存在性,再利用解的一致有界估计将局部解延拓至整体.我们主要的困难是扩散项的退化,为此,我们充分利用了分部积分公式、区域的有界性和Poincare不等式.另外,为了更清楚的研究前一模型中退化粘性项的扩散机制,我们还提出了这一方程的一个修改形式,即在非线性项上加了一个截断其低频部分的非局部算子的方程.我们考虑了这一带非局部算子方程的Cauchy问题,并得到了经典解的整体存在性和Lp衰减估计.
The nonlinear hyperbolic equations with dissipative mechanisms are very im-portant evolution equations. They play important roles in Mathematics, Physics,and many other fields. In this dissertation, we consider two types of nonlinear hy-perbolic equations with diferent dissipative mechanisms. One is the damped waveequation with nonlinear convection. The other is conservation law with degeneratedifusion term. We investigate the global existence and large time behavior of theclassical solutions to the two types of equations. This dissertation is arranged asfollows:
The first chapter introduces the dissertation. We first introduce backgroundinformation pertinent to the topic, and then review the Physics background andhistory of scientific studies on damped wave equations and degenerate hyperbolic-parabolic equations. Last, we give the specific problems we investigated in thisdissertation and summarize the main results we obtain.
In Chapter2, we study the global existence and Lp(2≤p≤∞) decay esti-mates of the classical solution to the Cauchy problem of the damped wave equationwith nonlinear convection term in the multi-dimensional space. We consider a smallperturbation around constant state u. Firstly, by using the Shizuta-Kawashimacondition which describes the dissipative mechanism of nonlinear hyperbolic equa-tions, we find that the dissipative structure of the equation depends on the constantstate u and the nonlinear term of the equation. Then, we show that when somedissipative condition holds, the classical solution to the Cauchy problem existsglobally. We employ the methods introduced by Ta-tsien Li ([31]). By using thedecay properties of the solution we get the global existence directly without provinglocal existence. Since the energy estimate method can only derive the bounded es-timates of the solution, we employ the frequency decomposition methods to obtain the decay properties of the solution. We decompose the solution into two parts:a low frequency part and a high frequency part. To deal with the low frequencypart, we use Green’s function. We use the estimates in the low frequency partof Green’s function in the linearized equation and Duhamel Principle to derivethe decay properties of the low frequency part of the solution. As for the highfrequency part, we use energy estimates and Poincare′-like inequality and get thedecay rate of solution. Lastly, the Lpestimates are obtained by using the a priorestimates, decomposition method, and interpolation lemma. Since we assume theinitial datum satisfies anti-derivatives conditions, the solution decays faster thanheat kernel.
In Chapter3, we continue to study the time-asymptotic behavior of the so-lution for the Cauchy problem of the damped wave equation with a nonlinearconvection term in multi-dimensional space. When the equation satisfies the dissi-pative condition, we obtained the pointwise decay estimates of the solution usingthe classical Green’s function methods. We accomplish this by firstly decomposingGreen’s function of linearized equations into three parts: low frequency, middlefrequency, and high frequency part and then derive pointwise estimates of Green’sfunction. Next, with the help of Duhamel Principle, we transform the nonlineardiferential equation into a nonlinear integral equation and get the expression ofthe solution. Then, we obtain the pointwise estimates of solutions by the a priorestimates. Given the derivative in the nonlinear term, we need to use energy es-timates to close the estimates on high derivative terms. Pointwise estimates canhelp us better understand the large time behavior of the solution. We find that thewave moves along a particular straight line and decays slowest along that straightline.
In Chapter4, we investigate the classical solution to the nonlinear degeneratehyperbolic-parabolic equations. We consider the initial-boundary value problem ofthe conservation law with a degenerate difusion term, and show that the classicalsolution exists globally and decays exponentially when t goes to infinity. To obtainthe global existence of the classical solution, we employ the standard continuityargument. Firstly, we prove the local solution exists and then extend it to a global solution by uniform estimates of the solution. The main difculty is the degenera-tion of the difusion term. To deal with it, we use integration by parts, the boundeddomain, and Poincare′inequality. Meanwhile, to clarify the viscous efect of the de-generate difusion term, we introduce a modified equation whose nonlinear term’slow frequency part was cut of by a nonlocal operator. We consider the Cauchyproblem of the equation with nonlocal operator and obtain global existence and Lpdecay estimates of classical solution.
第一章为绪论.我们首先介绍了课题背景,然后分别回顾了阻尼波动方程和退化双曲抛物方程的物理背景以及研究的历史和现状.最后,我们陈述了本文研究的具体问题及得到的主要结果.
在第二章,我们研究了多维空间中带非线性对流项的阻尼波动方程的Cauchy问题的经典解的整体存在性和Lp(2≤p≤(?))衰减估计.我们考虑方程在常状态u*附近的小扰动问题.首先,利用描述非线性发展方程耗散机制的Shizuta-Kawashima条件,我们发现方程的耗散结构由常状态u*和方程的非线性项共同决定.然后,我们证明了当方程满足某一耗散条件时,其Cauchy问题的整体经典解是存在的.本文采用了李大潜提出的整体迭代法([66]),不需要证明局部解的存在性,而是利用解的衰减性质直接得到了整体经典解的存在性.由于能量估计仅能得到解的有界估计,为得到解的衰减性质,我们使用了高低频分解的办法,将解分为两部分：低频部分和高频部分.对于低频部分,我们使用了Green函数办法.利用线性化方程的Green函数低频部分的估计以及Duhamel原理可以很自然地得到解的低频部分的衰减性质.对于高频部分,我们将能量估计和高频部分有最小频率的性质结合起来,利用Poincare-like不等式和Gronwall不等式也得到了解的衰减性质.最后,我们使用先验估计和频率分解方法得到了解的L2和L(?)的估计,并借助插值不等式得到了解的Lp估计.
在第三章中,我们继续研究多维空间中带非线性对流项的阻尼波动方程.在方程满足耗散条件的前提下,我们运用经典的Green函数方法得到了Cauchy问题解的逐点估计.首先将线性化方程的Green函数分为低频、带宽、高频三部分,再分别估计它们得到Green函数的逐点估计,然后运用Duhamel原理将非线性微分方程变为非线性积分方程,把解表达出来之后再利用先验估计的办法得到非线性问题解的逐点估计.因为非线性项含有解的导数,我们借助了能量估计来封闭高阶导数的估计.逐点估计可以让我们更清晰的看到解的大时间行为,我们发现,波的主体沿着某一条直线移动,并且沿着这条直线衰减最慢.
在第四章,我们试图对非线性退化双曲抛物方程的经典解问题进行研究.我们首先考虑了带退化扩散项的守恒律方程的初边值问题,我们证明了经典解是整体存在的,并且当时间趋于无穷大时解是指数衰减的.我们使用了经典的连续性办法来研究经典解的整体存在性.先证明局部解的存在性,再利用解的一致有界估计将局部解延拓至整体.我们主要的困难是扩散项的退化,为此,我们充分利用了分部积分公式、区域的有界性和Poincare不等式.另外,为了更清楚的研究前一模型中退化粘性项的扩散机制,我们还提出了这一方程的一个修改形式,即在非线性项上加了一个截断其低频部分的非局部算子的方程.我们考虑了这一带非局部算子方程的Cauchy问题,并得到了经典解的整体存在性和Lp衰减估计.
The nonlinear hyperbolic equations with dissipative mechanisms are very im-portant evolution equations. They play important roles in Mathematics, Physics,and many other fields. In this dissertation, we consider two types of nonlinear hy-perbolic equations with diferent dissipative mechanisms. One is the damped waveequation with nonlinear convection. The other is conservation law with degeneratedifusion term. We investigate the global existence and large time behavior of theclassical solutions to the two types of equations. This dissertation is arranged asfollows:
The first chapter introduces the dissertation. We first introduce backgroundinformation pertinent to the topic, and then review the Physics background andhistory of scientific studies on damped wave equations and degenerate hyperbolic-parabolic equations. Last, we give the specific problems we investigated in thisdissertation and summarize the main results we obtain.
In Chapter2, we study the global existence and Lp(2≤p≤∞) decay esti-mates of the classical solution to the Cauchy problem of the damped wave equationwith nonlinear convection term in the multi-dimensional space. We consider a smallperturbation around constant state u. Firstly, by using the Shizuta-Kawashimacondition which describes the dissipative mechanism of nonlinear hyperbolic equa-tions, we find that the dissipative structure of the equation depends on the constantstate u and the nonlinear term of the equation. Then, we show that when somedissipative condition holds, the classical solution to the Cauchy problem existsglobally. We employ the methods introduced by Ta-tsien Li ([31]). By using thedecay properties of the solution we get the global existence directly without provinglocal existence. Since the energy estimate method can only derive the bounded es-timates of the solution, we employ the frequency decomposition methods to obtain the decay properties of the solution. We decompose the solution into two parts:a low frequency part and a high frequency part. To deal with the low frequencypart, we use Green’s function. We use the estimates in the low frequency partof Green’s function in the linearized equation and Duhamel Principle to derivethe decay properties of the low frequency part of the solution. As for the highfrequency part, we use energy estimates and Poincare′-like inequality and get thedecay rate of solution. Lastly, the Lpestimates are obtained by using the a priorestimates, decomposition method, and interpolation lemma. Since we assume theinitial datum satisfies anti-derivatives conditions, the solution decays faster thanheat kernel.
In Chapter3, we continue to study the time-asymptotic behavior of the so-lution for the Cauchy problem of the damped wave equation with a nonlinearconvection term in multi-dimensional space. When the equation satisfies the dissi-pative condition, we obtained the pointwise decay estimates of the solution usingthe classical Green’s function methods. We accomplish this by firstly decomposingGreen’s function of linearized equations into three parts: low frequency, middlefrequency, and high frequency part and then derive pointwise estimates of Green’sfunction. Next, with the help of Duhamel Principle, we transform the nonlineardiferential equation into a nonlinear integral equation and get the expression ofthe solution. Then, we obtain the pointwise estimates of solutions by the a priorestimates. Given the derivative in the nonlinear term, we need to use energy es-timates to close the estimates on high derivative terms. Pointwise estimates canhelp us better understand the large time behavior of the solution. We find that thewave moves along a particular straight line and decays slowest along that straightline.
In Chapter4, we investigate the classical solution to the nonlinear degeneratehyperbolic-parabolic equations. We consider the initial-boundary value problem ofthe conservation law with a degenerate difusion term, and show that the classicalsolution exists globally and decays exponentially when t goes to infinity. To obtainthe global existence of the classical solution, we employ the standard continuityargument. Firstly, we prove the local solution exists and then extend it to a global solution by uniform estimates of the solution. The main difculty is the degenera-tion of the difusion term. To deal with it, we use integration by parts, the boundeddomain, and Poincare′inequality. Meanwhile, to clarify the viscous efect of the de-generate difusion term, we introduce a modified equation whose nonlinear term’slow frequency part was cut of by a nonlocal operator. We consider the Cauchyproblem of the equation with nonlocal operator and obtain global existence and Lpdecay estimates of classical solution.
引文
[1] M. C. Bustos, F. Concha, R. Bu¨rger, E. M. Tory, Sedimentation and Thicking: Phe-nomenological Foundation and Mathematical Theory, Kluwer Academic Publishers: Dor-drecht, Netherlands,1999.
[2] M. Bendahmane, K. H. Karlsen, Renormalized entropy solutions for quasilinearanisotropic degenerate parabolic equations, SIAM J. Math. Anal.36(2),405-422,2004.
[3] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech.Anal.147,269-361,1999.
[4] J. Chen, Global existence and Lpestimates for solutions of damped wave equation withnonlinear convection in multi-dimensional space, Acta Math. Sci. Ser. B Engl. Ed.32,no.3,1167-1180,2012.
[5] G.-Q. Chen and E. DiBenedetto, Stability of entropy solutions to the Cauchy problem fora class of nonlinear hyperbolic-parabolic equations, SIAM J. Math. Anal.33,751-762,2001.
[6] G. Q. Chen, K. H. Karlsen, Quasilinear anisotropic degenerate parabolic equations withtime-space dependent difusion coefcients, Comm. Pure Appl. Anal.4(2),241-266,2005.
[7] G. Q. Chen, K. H. Karlsen, L1framework for continuous dependence and error estimatesfor quasilinear degenerate parabolic equations, Trans. Amer. Math. Soc.,(3)358,937-963,2005.
[8] J. Chen, Y. Li, W. Wang, Global existence of solutions to the initial-boundary valueproblem of conservation law with degenerate difusion term, Nonlinear Anal.,78,47-61,2013.
[9]G. Q. Chen, B. Perthame, Well-posedness for nonisotropic degenerate parabolichyperbolic equations, Annales de1'Institut Henri Poincar'e Analyse Non Lin'eaire,20,645-668,2003.
[10]G. Q. Chen, B. Perthame, Large-time behavior of periodic entropy solutions to anisotropic degenerate parabolic-hyperbolic equations, Proc. Amer. Math. Soc.137(9),3003-3011,2009.
[11]J. Chen, W. Wang, The point-wise estimates for the solutions of damped wave equation with nonlinear convection in multi-dimensional space, accepted by Comm. Pure Appl. Anal.
[12]S. J. Deng, W. K. Wang, Pointwise Decaying Rate of Large Pertuibation Around Viscous Shock for Scalar Viscous Conservation Law,, Unpublished results.
[13]S. J. Deng, W. K. Wang, S. H. Yu, Pointwise convergence to Knudsen layers of the Boltzmann equation, Comm. Math. Phys.,281,287-347,2008.
[14]L. C. Evans, Partial Differential Equations, Graduate in Math.,19, Amer. Math. Soc., Providence, RI,1998.
[15]L L Fan., H X Liu, H. Yin, Dacay estimates of planar stationary waves for damped wave equations with nonlinear convection in mutil-dimensional half space. Acta Math Sci,31(B)(4),1389-1410,2011.
[16]J. Goodman, Stability of Viscous Scalar Shock Fronts in Several Dimensions, Trans. Amer. Math. Soc.,311,683-695,1989.
[17]J. Goodman, J. R. Miller, Long-Time Behavior of Scalar Viscous Shock Fronts in Two Dimensions, J. Dynamics and Differential Equations,11(2),255-277,1999.
[18]T. Hosono, T. Ogawa, Large time behavior and Lp-Lq estimate of solutions of2-dimensional nonlinear damped wave equations, J. Differ. Equations,203,82-118,2004.
[19]D. Hoff, K. Zumbrum, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys.,48,597-614,1997.
[20]R. Ikehata, A remark on a critical exponent for the semilinear dissipative wave equation in the one dimensional half space, Differential Integral Equations,16,727-736,2003.
[21]R. Ikehata, K. Nishihara, H. J. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differ. Equation,226,1-29,2006.
[22]L. V. Juan, The porous medium equation:mathematical theory, Oxford university press,2006.
[23]D.B. Kotlow, Qusilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data, J. Math. Anal. Appl.35,563-576,1971.
[24]S. Kawashima, A. Matsumura, Asymptotic Stability of Traveling Wave Solutions of Sys-tems for One-dimensional Gas Motion, Conim. Math. Phys.101,97-127,1985.
[25]S. Kawashima, M. Nakao, K. Ono,On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan47,617-653,1995.
[26]K. H. Karlsen, N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coe+dents, Discrete Contin. Dyn. Syst.,9(5),1081-1104,2003.
[27]K. H. Karlsen and M. Ohlberger, A note on the uniqueness of entropy solutions of non-linear degenerate parabolic equations, J. Math. Anal. Appl.,275(1),439-458,2002.
[28]D. L. Li, The Green's function of the Navier-Stokes equations for gas dynamics in R3, Commmun. Math. Phys.257,579-619,2005.
[29] T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm.Pure Appl. Math,50,1113-1182,1997.
[30] Y. Q. Liu, The point-wise estimates of solutions for semi-linear dissipative wave equation,Comm. Pure Appl. Anal.,12,237-252,2013.
[31] Ta-tsien Li, Y. M. Chen, Global Classical Solutions for Nonlinear Evolution Equations,Pitman Monogr. Surv. Pure Appl. Math., vol.45, Longman Scientific and Technical,Harlow,1992.
[32] Y. Q. Liu, S. Kawashima, Decay property for the Timoshenko system with memory-typedissipation, Math. Models Methods Appl. Sci.,22, no.2,1150012,19pp,2012.
[33] Y. Q. Liu, S. Kawashima, Global existence and asymptotic decay of solutions to the non-linear Timoshenko system with memory, Nonlinear Anal.,84,1-17,2013.
[34] J. Lin, K. Nishihara, J. Zhai, L2-estimates of solutions for damped wave equationswithspace-time dependent damping term, J. Diferential Equations,248,403-422,2010.
[35] T.-P. Liu, W. K. Wang, The pointwise estimates of difusion wave for the Navier-Stokessystems in odd multi-dimensions, Comm. Math. Phys.,169,145-173,1998.
[36] Y. C. Li, Q. Wang, Homogeneous Dirichlet problems for quasilinear anisotropic degenerateparabolic-hyperbolic equations, J. Diferential Equations,252(9),4719-4741,2012.
[37] Y. Q. Liu, W. K. Wang, The pointwise estimates of solutions for dissipative wave equationin multi-dimensions, Discrete Contin. Dyn. Syst.,20,1013-1028,2008.
[38] T.-P. Liu, Y. Zeng, Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws, A. M. S. memoirs,599,1997.
[39] A. Matsumura, On the asymptotic behavior of solutions or semi-linear wave equations.Publ. Res. Inst. Math. Sci.,12,169-189,1976.
[40] P. Marchti, K. Nishihara, The LP Lqestimates of solutions to one-dimensional dampedwave eqautions and their application to the compressible flow through porous media.J.Diferential Equations,191,445-469,2003.
[41] K. Nishihara, Lp Lqestimates of solutions to the damped wave equation in3-dimensionalspace and their application. Math. Z.,244,631-649,2003.
[42] K. Nishihara, Global asymptotics for the damped wace equation with absotption in higherdimensional space, J. Math. Soc. Japan,58(3),805-836,2006.
[43] M. Nakao, K. Ono, Existence of global solutions to the Cauchy problem for the semilieardissipative wave equation, Math. Z.,214,325-241,1993.
[44] K. Nishihara, H. J. Zhao,Dacay properties of solutions to the Cauchy problem for thedamped wace equation with absorption, J. Math. Anal. Appl.,313,698-610,2006.
[45] K. Nishihara, J. Zhai, Asymptotic behaviors of solutions for time dependent damped waveequations, J. Math. Anal. Appl.,360,412-421,2009.
[46] K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissi-pative wave equations, Discrete Contin. Dyn. Syst.,9, no.3,651-662,2003.
[47] K. Ono, Lpdecay problem for the dissipative wave equation in even dimensions. Math.Meth. Appl. Sci.,27,1843-1863,2004.
[48] K. Ono, Lpdecay problem for the dissipative wave equation in odd dimensions. J. Math.Anal. Appl.,310,347-361,2005.
[49] B. Perthame, P. E. Souganidis, Dissipative and entropy solutions to non-isotropic degen-erate parabolic balance laws, Arch. Rational Mech Anal.,170(4),359-370,2003.
[50] J. Smoller, Shock Waves and Reaction-Difusion Equations, Springer-Verlag, Berlin,1994.
[51] Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic-parabolic type with appli-cations to the discrete Boltaman Equation, Hokkaido Math. J.,14,249-275,1985.
[52] Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with anonlinear convection term, Adv. Math. Sci. Appl.,18, no.1,329-343,2008.
[53] Y. Ueda, T. Nakamura, S. Kawashima, Stability of planar stationary waves for dampedwave equations with nonlinear convection in multi-dimensional half space, Kinet. Relat.Models1, no.1,49-64,2008.
[54] Volpert, A. I., Hugjaev, S. I., Cauchy,s problem for degenerate second order quasilinearparabolic equation, Transl. Math. USSR Sb7,365-387,1969.
[55] W. J. Wang, Decay property of regularity-loss type for nonlinear hyperbolic-elliptic systemin multi-dimensions, J. Math. Anal. Appl.,387, no.2,993-1008,2012.
[56] Z. G. Wang, Y. C. Li, The homogeneous Drichlet problem for quasilinear anisotropic de-generate parabolic-hyperbolic equations with Lpinitial value, Acta Mathematica Scientia,5,1727-1742,2012.
[57] W. J. Wang, W. K. Wang, The pointwise estimates of solutions for semilinear dissipativewave equation in multi-dimensions, J. Math. Anal. Appl.,366,226-241,2010.
[58] W. K. Wang, W. J. Wang, The pointwise estimates of solutions for a model system ofthe radiating gas in multi-dimensions, Nonlinear Anal.,71,1180-1195,2009.
[59] W. K. Wang, Z. G. Wu, Pointwise estimate of solutions for the Navier-Stokes-Poissonequation in multi-dimensions, Journal Diferential Equations,248,1617-1636,2010.
[60] Z. G. Wang, L. Wang, Y. C. Li, Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefcients, Commun. Pure Appl. Anal.,12(3),1163-1182,2013.
[61] W. K. Wang, T. Yang, The pointwise estimates of solutions for Euler equations withdamping in multi-dimensions, J. Difer. Equations,173,410-450,2001.
[62]Z. Wu, J. Yin, Some properties of functions in BVx and their applications to the unique-ness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J.,5(4),395-422,1989.
[63]Y. Zeng, Gas Dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Rational Mech. Anal.,150,225-279,1999.
[64]Y. Zeng, Gas flows with several thermal nonequilibrium modes, Arch Rathinal Mech Anal,196,191-225,2010.
[65]Y. Zeng, Large time behavior of solutions to nonlinear viscoelastic model with fading memory, Acta Mathematica Scientia,32B(1),219-236,2012.
[66]李大潜,陈韵梅,非线性发展方程.北京：科学出版社,1989.
[67]李亚纯,具有耗散项的完全非线性波动方程的经典解,数学年刊,4,451-466,1996.
[68]周民强,调和分析讲义(实变方法),北京大学出版社,2003.
[2] M. Bendahmane, K. H. Karlsen, Renormalized entropy solutions for quasilinearanisotropic degenerate parabolic equations, SIAM J. Math. Anal.36(2),405-422,2004.
[3] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Rational Mech.Anal.147,269-361,1999.
[4] J. Chen, Global existence and Lpestimates for solutions of damped wave equation withnonlinear convection in multi-dimensional space, Acta Math. Sci. Ser. B Engl. Ed.32,no.3,1167-1180,2012.
[5] G.-Q. Chen and E. DiBenedetto, Stability of entropy solutions to the Cauchy problem fora class of nonlinear hyperbolic-parabolic equations, SIAM J. Math. Anal.33,751-762,2001.
[6] G. Q. Chen, K. H. Karlsen, Quasilinear anisotropic degenerate parabolic equations withtime-space dependent difusion coefcients, Comm. Pure Appl. Anal.4(2),241-266,2005.
[7] G. Q. Chen, K. H. Karlsen, L1framework for continuous dependence and error estimatesfor quasilinear degenerate parabolic equations, Trans. Amer. Math. Soc.,(3)358,937-963,2005.
[8] J. Chen, Y. Li, W. Wang, Global existence of solutions to the initial-boundary valueproblem of conservation law with degenerate difusion term, Nonlinear Anal.,78,47-61,2013.
[9]G. Q. Chen, B. Perthame, Well-posedness for nonisotropic degenerate parabolichyperbolic equations, Annales de1'Institut Henri Poincar'e Analyse Non Lin'eaire,20,645-668,2003.
[10]G. Q. Chen, B. Perthame, Large-time behavior of periodic entropy solutions to anisotropic degenerate parabolic-hyperbolic equations, Proc. Amer. Math. Soc.137(9),3003-3011,2009.
[11]J. Chen, W. Wang, The point-wise estimates for the solutions of damped wave equation with nonlinear convection in multi-dimensional space, accepted by Comm. Pure Appl. Anal.
[12]S. J. Deng, W. K. Wang, Pointwise Decaying Rate of Large Pertuibation Around Viscous Shock for Scalar Viscous Conservation Law,, Unpublished results.
[13]S. J. Deng, W. K. Wang, S. H. Yu, Pointwise convergence to Knudsen layers of the Boltzmann equation, Comm. Math. Phys.,281,287-347,2008.
[14]L. C. Evans, Partial Differential Equations, Graduate in Math.,19, Amer. Math. Soc., Providence, RI,1998.
[15]L L Fan., H X Liu, H. Yin, Dacay estimates of planar stationary waves for damped wave equations with nonlinear convection in mutil-dimensional half space. Acta Math Sci,31(B)(4),1389-1410,2011.
[16]J. Goodman, Stability of Viscous Scalar Shock Fronts in Several Dimensions, Trans. Amer. Math. Soc.,311,683-695,1989.
[17]J. Goodman, J. R. Miller, Long-Time Behavior of Scalar Viscous Shock Fronts in Two Dimensions, J. Dynamics and Differential Equations,11(2),255-277,1999.
[18]T. Hosono, T. Ogawa, Large time behavior and Lp-Lq estimate of solutions of2-dimensional nonlinear damped wave equations, J. Differ. Equations,203,82-118,2004.
[19]D. Hoff, K. Zumbrum, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys.,48,597-614,1997.
[20]R. Ikehata, A remark on a critical exponent for the semilinear dissipative wave equation in the one dimensional half space, Differential Integral Equations,16,727-736,2003.
[21]R. Ikehata, K. Nishihara, H. J. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differ. Equation,226,1-29,2006.
[22]L. V. Juan, The porous medium equation:mathematical theory, Oxford university press,2006.
[23]D.B. Kotlow, Qusilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data, J. Math. Anal. Appl.35,563-576,1971.
[24]S. Kawashima, A. Matsumura, Asymptotic Stability of Traveling Wave Solutions of Sys-tems for One-dimensional Gas Motion, Conim. Math. Phys.101,97-127,1985.
[25]S. Kawashima, M. Nakao, K. Ono,On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan47,617-653,1995.
[26]K. H. Karlsen, N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coe+dents, Discrete Contin. Dyn. Syst.,9(5),1081-1104,2003.
[27]K. H. Karlsen and M. Ohlberger, A note on the uniqueness of entropy solutions of non-linear degenerate parabolic equations, J. Math. Anal. Appl.,275(1),439-458,2002.
[28]D. L. Li, The Green's function of the Navier-Stokes equations for gas dynamics in R3, Commmun. Math. Phys.257,579-619,2005.
[29] T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm.Pure Appl. Math,50,1113-1182,1997.
[30] Y. Q. Liu, The point-wise estimates of solutions for semi-linear dissipative wave equation,Comm. Pure Appl. Anal.,12,237-252,2013.
[31] Ta-tsien Li, Y. M. Chen, Global Classical Solutions for Nonlinear Evolution Equations,Pitman Monogr. Surv. Pure Appl. Math., vol.45, Longman Scientific and Technical,Harlow,1992.
[32] Y. Q. Liu, S. Kawashima, Decay property for the Timoshenko system with memory-typedissipation, Math. Models Methods Appl. Sci.,22, no.2,1150012,19pp,2012.
[33] Y. Q. Liu, S. Kawashima, Global existence and asymptotic decay of solutions to the non-linear Timoshenko system with memory, Nonlinear Anal.,84,1-17,2013.
[34] J. Lin, K. Nishihara, J. Zhai, L2-estimates of solutions for damped wave equationswithspace-time dependent damping term, J. Diferential Equations,248,403-422,2010.
[35] T.-P. Liu, W. K. Wang, The pointwise estimates of difusion wave for the Navier-Stokessystems in odd multi-dimensions, Comm. Math. Phys.,169,145-173,1998.
[36] Y. C. Li, Q. Wang, Homogeneous Dirichlet problems for quasilinear anisotropic degenerateparabolic-hyperbolic equations, J. Diferential Equations,252(9),4719-4741,2012.
[37] Y. Q. Liu, W. K. Wang, The pointwise estimates of solutions for dissipative wave equationin multi-dimensions, Discrete Contin. Dyn. Syst.,20,1013-1028,2008.
[38] T.-P. Liu, Y. Zeng, Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws, A. M. S. memoirs,599,1997.
[39] A. Matsumura, On the asymptotic behavior of solutions or semi-linear wave equations.Publ. Res. Inst. Math. Sci.,12,169-189,1976.
[40] P. Marchti, K. Nishihara, The LP Lqestimates of solutions to one-dimensional dampedwave eqautions and their application to the compressible flow through porous media.J.Diferential Equations,191,445-469,2003.
[41] K. Nishihara, Lp Lqestimates of solutions to the damped wave equation in3-dimensionalspace and their application. Math. Z.,244,631-649,2003.
[42] K. Nishihara, Global asymptotics for the damped wace equation with absotption in higherdimensional space, J. Math. Soc. Japan,58(3),805-836,2006.
[43] M. Nakao, K. Ono, Existence of global solutions to the Cauchy problem for the semilieardissipative wave equation, Math. Z.,214,325-241,1993.
[44] K. Nishihara, H. J. Zhao,Dacay properties of solutions to the Cauchy problem for thedamped wace equation with absorption, J. Math. Anal. Appl.,313,698-610,2006.
[45] K. Nishihara, J. Zhai, Asymptotic behaviors of solutions for time dependent damped waveequations, J. Math. Anal. Appl.,360,412-421,2009.
[46] K. Ono, Global existence and asymptotic behavior of small solutions for semilinear dissi-pative wave equations, Discrete Contin. Dyn. Syst.,9, no.3,651-662,2003.
[47] K. Ono, Lpdecay problem for the dissipative wave equation in even dimensions. Math.Meth. Appl. Sci.,27,1843-1863,2004.
[48] K. Ono, Lpdecay problem for the dissipative wave equation in odd dimensions. J. Math.Anal. Appl.,310,347-361,2005.
[49] B. Perthame, P. E. Souganidis, Dissipative and entropy solutions to non-isotropic degen-erate parabolic balance laws, Arch. Rational Mech Anal.,170(4),359-370,2003.
[50] J. Smoller, Shock Waves and Reaction-Difusion Equations, Springer-Verlag, Berlin,1994.
[51] Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic-parabolic type with appli-cations to the discrete Boltaman Equation, Hokkaido Math. J.,14,249-275,1985.
[52] Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with anonlinear convection term, Adv. Math. Sci. Appl.,18, no.1,329-343,2008.
[53] Y. Ueda, T. Nakamura, S. Kawashima, Stability of planar stationary waves for dampedwave equations with nonlinear convection in multi-dimensional half space, Kinet. Relat.Models1, no.1,49-64,2008.
[54] Volpert, A. I., Hugjaev, S. I., Cauchy,s problem for degenerate second order quasilinearparabolic equation, Transl. Math. USSR Sb7,365-387,1969.
[55] W. J. Wang, Decay property of regularity-loss type for nonlinear hyperbolic-elliptic systemin multi-dimensions, J. Math. Anal. Appl.,387, no.2,993-1008,2012.
[56] Z. G. Wang, Y. C. Li, The homogeneous Drichlet problem for quasilinear anisotropic de-generate parabolic-hyperbolic equations with Lpinitial value, Acta Mathematica Scientia,5,1727-1742,2012.
[57] W. J. Wang, W. K. Wang, The pointwise estimates of solutions for semilinear dissipativewave equation in multi-dimensions, J. Math. Anal. Appl.,366,226-241,2010.
[58] W. K. Wang, W. J. Wang, The pointwise estimates of solutions for a model system ofthe radiating gas in multi-dimensions, Nonlinear Anal.,71,1180-1195,2009.
[59] W. K. Wang, Z. G. Wu, Pointwise estimate of solutions for the Navier-Stokes-Poissonequation in multi-dimensions, Journal Diferential Equations,248,1617-1636,2010.
[60] Z. G. Wang, L. Wang, Y. C. Li, Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefcients, Commun. Pure Appl. Anal.,12(3),1163-1182,2013.
[61] W. K. Wang, T. Yang, The pointwise estimates of solutions for Euler equations withdamping in multi-dimensions, J. Difer. Equations,173,410-450,2001.
[62]Z. Wu, J. Yin, Some properties of functions in BVx and their applications to the unique-ness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J.,5(4),395-422,1989.
[63]Y. Zeng, Gas Dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Rational Mech. Anal.,150,225-279,1999.
[64]Y. Zeng, Gas flows with several thermal nonequilibrium modes, Arch Rathinal Mech Anal,196,191-225,2010.
[65]Y. Zeng, Large time behavior of solutions to nonlinear viscoelastic model with fading memory, Acta Mathematica Scientia,32B(1),219-236,2012.
[66]李大潜,陈韵梅,非线性发展方程.北京：科学出版社,1989.
[67]李亚纯,具有耗散项的完全非线性波动方程的经典解,数学年刊,4,451-466,1996.
[68]周民强,调和分析讲义(实变方法),北京大学出版社,2003.