帶有記憶項(xiàng)的板方程的解的衰減估計(jì)研究
發(fā)布時(shí)間：2018-08-05 15:30
【摘要】：數(shù)學(xué)研究中偏微分方程的應(yīng)用和物理等學(xué)科緊密聯(lián)系在一起,相互推動(dòng)、促進(jìn),而非線性偏微分方程的研究已經(jīng)成為研究的重要課題之一。對于耗散性偏微分方程(組)的研究已有比較長的歷史,結(jié)果也比較豐富。近年來,對于損失正則性類型的偏微分方程受到了國內(nèi)外的廣泛關(guān)注和研究。梁、板等的振動(dòng)所滿足的偏微分方程就屬于正則性損失類型。目前,已成為一個(gè)比較活躍的基本研究課題。本論文主要研究此類帶有摩擦項(xiàng)及記憶項(xiàng)的半線性板方程(即,帶有時(shí)間延遲項(xiàng)的板方程)的解的衰減估計(jì)及正則性損失問題。我們首先研究線性部分所對應(yīng)的方程。通過傅立葉變換將問題轉(zhuǎn)化為頻率空間中的估計(jì)。利用常微分方程及能量估計(jì)方法得到頻率空間中基本解算子的逐點(diǎn)估計(jì)。由于方程的記憶項(xiàng)中含有關(guān)于時(shí)間的偏導(dǎo)數(shù),傳統(tǒng)文獻(xiàn)中的方法不能直接應(yīng)用。為了解決這個(gè)問題,我們將方程轉(zhuǎn)化為一類具有特殊形式的非齊次線性方程,對非齊次項(xiàng)進(jìn)行估計(jì),得出線性部分所對應(yīng)的方程解的衰減估計(jì)與正則性損失的關(guān)系。然后對半線性方程,我們對Sobolev空間進(jìn)行時(shí)間加權(quán),構(gòu)造一類時(shí)間加權(quán)范數(shù),應(yīng)用壓縮映射定理的不動(dòng)點(diǎn)原理得到半線性問題解的全局存在性及時(shí)間衰減估計(jì)。本論文的創(chuàng)新之處在于:與已有的傳統(tǒng)文獻(xiàn)的結(jié)果相比,本論文所研究的這類方程的解的衰減估計(jì)和正則性損失都是由高頻部分決定的。不需要對初值作1(?9))假設(shè),就可以得到相同類型的結(jié)論。我們通過對帶有記憶項(xiàng)的板方程解的衰減性及正則性的研究,得到了一些比較有意義的結(jié)果,總結(jié)了對一類板方程解的衰減性質(zhì)的各種估計(jì)的方法,這對我們以后的研究有很大的幫助。希望以后可以把我們的研究結(jié)果及方法用到其他一些相關(guān)的問題的研究中去。我們將結(jié)合多種類型的板方程,針對它們的特點(diǎn),尋找各種適當(dāng)?shù)暮啙嵉难芯糠椒?盡量在方法上有新的突破,并得到解的最優(yōu)衰減估計(jì)。
[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.
【學(xué)位授予單位】：華北電力大學(xué)(北京)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.2
本文編號：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.
【學(xué)位授予單位】：華北電力大學(xué)(北京)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.2
【參考文獻(xiàn)】
相關(guān)博士學(xué)位論文 前1條
1 武潔瓊;若干非線性波動(dòng)方程的解的性質(zhì)和控制問題[D];山西大學(xué);2010年
相關(guān)碩士學(xué)位論文 前2條
1 蘭杰;兩類偏微分方程的解的性質(zhì)研究[D];山西大學(xué);2015年
2 榮曉亮;一類非線性熱彈板方程的初邊值問題[D];太原理工大學(xué);2008年
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