带有记忆项的板方程的解的衰减估计研究
发布时间：2018-08-05 15:30
【摘要】：数学研究中偏微分方程的应用和物理等学科紧密联系在一起,相互推动、促进,而非线性偏微分方程的研究已经成为研究的重要课题之一。对于耗散性偏微分方程(组)的研究已有比较长的历史,结果也比较丰富。近年来,对于损失正则性类型的偏微分方程受到了国内外的广泛关注和研究。梁、板等的振动所满足的偏微分方程就属于正则性损失类型。目前,已成为一个比较活跃的基本研究课题。本论文主要研究此类带有摩擦项及记忆项的半线性板方程(即,带有时间延迟项的板方程)的解的衰减估计及正则性损失问题。我们首先研究线性部分所对应的方程。通过傅立叶变换将问题转化为频率空间中的估计。利用常微分方程及能量估计方法得到频率空间中基本解算子的逐点估计。由于方程的记忆项中含有关于时间的偏导数,传统文献中的方法不能直接应用。为了解决这个问题,我们将方程转化为一类具有特殊形式的非齐次线性方程,对非齐次项进行估计,得出线性部分所对应的方程解的衰减估计与正则性损失的关系。然后对半线性方程,我们对Sobolev空间进行时间加权,构造一类时间加权范数,应用压缩映射定理的不动点原理得到半线性问题解的全局存在性及时间衰减估计。本论文的创新之处在于:与已有的传统文献的结果相比,本论文所研究的这类方程的解的衰减估计和正则性损失都是由高频部分决定的。不需要对初值作1(?9))假设,就可以得到相同类型的结论。我们通过对带有记忆项的板方程解的衰减性及正则性的研究,得到了一些比较有意义的结果,总结了对一类板方程解的衰减性质的各种估计的方法,这对我们以后的研究有很大的帮助。希望以后可以把我们的研究结果及方法用到其他一些相关的问题的研究中去。我们将结合多种类型的板方程,针对它们的特点,寻找各种适当的简洁的研究方法,尽量在方法上有新的突破,并得到解的最优衰减估计。
[Abstract]:The application of partial differential equations (PDEs) in mathematical research is closely linked with physics and so on. The study of nonlinear PDEs has become one of the important research topics. The study of dissipative partial differential equations has a long history and rich results. In recent years, partial differential equations of loss regularity type have received extensive attention and research at home and abroad. The partial differential equation satisfied by the vibration of beam, plate and so on belongs to the regular loss type. At present, has become a relatively active basic research topic. In this paper, we study the decay estimation and regularity loss of the solution of the semilinear plate equation with friction term and memory term (that is, the plate equation with time delay term). We first study the equation corresponding to the linear part. The problem is transformed into the estimation in frequency space by Fourier transform. By using ordinary differential equation and energy estimation method, the point by point estimator of fundamental solution operator in frequency space is obtained. Because the memory term of the equation contains partial derivatives of time, the traditional methods can not be applied directly. In order to solve this problem, we transform the equation into a class of inhomogeneous linear equations with special forms, estimate the nonhomogeneous terms, and obtain the relationship between the decay estimate of the solution of the equation corresponding to the linear part and the loss of regularity. Then we construct a class of time-weighted norm for semilinear equations by time weighting on Sobolev space. By applying the fixed point principle of contraction mapping theorem, we obtain the global existence and time decay estimates of the solutions of semilinear problems. The innovation of this paper is that the attenuation estimation and regularity loss of the solutions of the equations studied in this paper are determined by the high frequency part compared with the results of the traditional literature. The same type of conclusion can be obtained without the assumption of 1 (9) for the initial value. By studying the attenuation and regularity of the solutions of a class of plate equations with memory terms, we obtain some meaningful results and summarize various methods for estimating the decay properties of solutions of a class of plate equations. This will be of great help to our future research. We hope to apply our research results and methods to some other related problems in the future. According to the characteristics of various kinds of plate equations, we will search for a variety of suitable and succinct research methods, try our best to make a new breakthrough in the method, and obtain the optimal decay estimate of the solution.
【学位授予单位】：华北电力大学(北京)
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.2
本文编号：2166219
[Abstract]:The application of partial differential equations (PDEs) in mathematical research is closely linked with physics and so on. The study of nonlinear PDEs has become one of the important research topics. The study of dissipative partial differential equations has a long history and rich results. In recent years, partial differential equations of loss regularity type have received extensive attention and research at home and abroad. The partial differential equation satisfied by the vibration of beam, plate and so on belongs to the regular loss type. At present, has become a relatively active basic research topic. In this paper, we study the decay estimation and regularity loss of the solution of the semilinear plate equation with friction term and memory term (that is, the plate equation with time delay term). We first study the equation corresponding to the linear part. The problem is transformed into the estimation in frequency space by Fourier transform. By using ordinary differential equation and energy estimation method, the point by point estimator of fundamental solution operator in frequency space is obtained. Because the memory term of the equation contains partial derivatives of time, the traditional methods can not be applied directly. In order to solve this problem, we transform the equation into a class of inhomogeneous linear equations with special forms, estimate the nonhomogeneous terms, and obtain the relationship between the decay estimate of the solution of the equation corresponding to the linear part and the loss of regularity. Then we construct a class of time-weighted norm for semilinear equations by time weighting on Sobolev space. By applying the fixed point principle of contraction mapping theorem, we obtain the global existence and time decay estimates of the solutions of semilinear problems. The innovation of this paper is that the attenuation estimation and regularity loss of the solutions of the equations studied in this paper are determined by the high frequency part compared with the results of the traditional literature. The same type of conclusion can be obtained without the assumption of 1 (9) for the initial value. By studying the attenuation and regularity of the solutions of a class of plate equations with memory terms, we obtain some meaningful results and summarize various methods for estimating the decay properties of solutions of a class of plate equations. This will be of great help to our future research. We hope to apply our research results and methods to some other related problems in the future. According to the characteristics of various kinds of plate equations, we will search for a variety of suitable and succinct research methods, try our best to make a new breakthrough in the method, and obtain the optimal decay estimate of the solution.
【学位授予单位】：华北电力大学(北京)
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.2
【参考文献】
相关博士学位论文 前1条
1 武洁琼;若干非线性波动方程的解的性质和控制问题[D];山西大学;2010年
相关硕士学位论文 前2条
1 兰杰;两类偏微分方程的解的性质研究[D];山西大学;2015年
2 荣晓亮;一类非线性热弹板方程的初边值问题[D];太原理工大学;2008年
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