【摘要】：本文主要研究了分數(shù)階長短波方程,(2+1)維分數(shù)階長短波方程的長時間行為,得到了廣義非自治分數(shù)階長短波方程周期初邊值問題整體光滑解的存在唯一性,廣義非自治分數(shù)階長短波方程一致吸引子的存在性,并且證明了(2+1)維分數(shù)階長短波方程周期初邊值問題存在唯一整體光滑解以及(2+1)維分數(shù)階長短波方程整體吸引子的存在性.本文共分為五個部分.第一部分,主要闡述了分數(shù)階微積分,無窮維動力系統(tǒng)以及長短波方程的物理背景及相關(guān)理論知識,回顧已有的部分研究成果,最后簡單介紹了一下本文的主要研究工作.第二部分,考慮了廣義分數(shù)階長短波方程的周期初邊值問題.首先運用Gagliardo-Nirenberg不等式,Young不等式和Gronwall不等式進行一致先驗估計,其次利用Gal?rkin方法,研究了廣義非自治分數(shù)階長短波方程的周期初邊值問題光滑解的整體存在性和唯一性.第三部分,證明了廣義分數(shù)階非自治長短波方程一致吸引子的存在性.首先通過一致先驗估計以及Gal?rkin方法即可證明該周期初邊值問題解的存在唯一性,其次利用非自治系統(tǒng)一致吸引子的有關(guān)理論得到了該方程強緊一致吸引子的存在性.第四部分,考慮了(2+1)維分數(shù)階長短波方程的周期初邊值問題.利用一致先驗估計和Gal?rkin方法證明了該方程周期初邊值問題整體光滑解的存在性.第五部分,證明了(2+1)維分數(shù)階長短波方程整體吸引子的存在性.結(jié)合解半群性質(zhì),運用弱收斂方法證明了(2+1)維分數(shù)階長短波方程的周期初邊值問題整體強吸引子存在.
[Abstract]:In this paper, we mainly study the long-time behavior of fractional-order long-wave equation and (21)-dimensional fractional-order long-wave equation, and obtain the existence and uniqueness of global smooth solution for the periodic initial-boundary value problem of generalized non-autonomous fractional-order long-wave equation. Existence of uniform Attractors for Generalized non-autonomous Fractional order long short Wave equations, It is also proved that there exists a unique global smooth solution for the periodic initial boundary value problem of the (21) dimensional fractional order long short wave equation and the existence of the global attractor for the (21) dimensional fractional order long short wave equation. This paper is divided into five parts. In the first part, the physical background and related theoretical knowledge of fractional calculus, infinite-dimensional dynamical system and long-short wave equation are described, and the existing research results are reviewed. Finally, the main research work of this paper is briefly introduced. In the second part, we consider the periodic initial-boundary value problem of the generalized fractional order short-wave equation. In this paper, we first use Gagliardo-Nirenberg inequality, Young inequality and Gronwall inequality to obtain uniform prior estimates. Secondly, we use Gal?rkin method to study the global existence and uniqueness of smooth solutions to the periodic initial boundary value problems of generalized fractional order short wave equations. In the third part, the existence of uniform attractors for generalized fractional order non-autonomous short-wave equations is proved. First, the existence and uniqueness of the solution of the periodic initial-boundary value problem can be proved by using the uniform prior estimate and the Gal?rkin method. Secondly, the existence of the strongly compact uniform attractor for the equation is obtained by using the theory of the uniform attractor of the non-autonomous system. In the fourth part, the periodic initial boundary value problem of the (21) dimensional fractional order long short wave equation is considered. The existence of global smooth solutions to the periodic initial boundary value problem of the equation is proved by means of uniform prior estimates and Gal?rkin 's method. In the fifth part, the existence of the global attractor for the (21) dimensional fractional order short wave equation is proved. In this paper, the existence of global strong attractors for the periodic initial boundary value problem of (21) dimensional fractional order short wave equation is proved by using the weak convergence method combined with the properties of the solution semigroup.
【學位授予單位】：魯東大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O175

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本文編號：2462186