發(fā)布時間：2018-06-21 09:45

  本文選題：非局部擴散 + 局部化源項�。� 參考：《東南大學(xué)》2015年碩士論文

【摘要】：非線性偏微分方程(組)解的性質(zhì)一直以來都是非線性分析和偏微分方程這兩個研究領(lǐng)域討論的一個重要內(nèi)容.生物學(xué)、化學(xué)和物理學(xué)等應(yīng)用學(xué)科中的很多數(shù)學(xué)模型也都與這些方程緊密相關(guān).隨著科學(xué)技術(shù)的日新月異和數(shù)學(xué)研究方法的日臻完善,非線性偏微分方程(組)的形式越來越多樣.近年來,非局部擴散方程以及推廣而來的一系列非局部擴散問題,引起了很多科學(xué)工作者的興趣和關(guān)注.這些非局部擴散問題被廣泛地用來描述擴散的進程,其中u(x,t)可以用來表示某個物種在t時刻在點x處的密度,J(x-y)代表從點y移動到點x的概率分布,卷積(J+u)(x,t)=∫RN J(x-y)u(y,t)dy表示物種從其它點到達點x的速度.本論文研究如下帶有局部化源項的非局部擴散方程組解的性質(zhì),包括解的存在性與唯一性,解的整體存在與有限時刻爆破,解的兩個分量u和v發(fā)生同時爆破和不同時爆破的條件以及解的爆破速率的估計等問題.在本文中,我們首先運用壓縮映像原理證明了解的存在性和唯一性；然后通過建立新的比較原理,并利用上下解方法導(dǎo)出了解在有限時刻發(fā)生爆破的條件；接著利用一些常用的不等式和分析的技巧并借鑒文獻[1]中的方法,我們得到：u和v在有限時刻T同時爆破的充分條件和必要條件,并進一步討論了u和u的爆破模式與爆破點集的刻畫；最后借助于比較原理和常微分方程不等式等方法和技巧,導(dǎo)出了u和v在有限時刻T不同時爆破的充分條件和必要條件以及爆破速率的估計和爆破點集的刻畫.
[Abstract]:The properties of solutions of nonlinear partial differential equations have always been an important part of nonlinear analysis and partial differential equations. Many mathematical models in applied disciplines such as biology, chemistry and physics are also closely related to these equations. With the rapid development of science and technology and the improvement of mathematical research methods, the forms of nonlinear partial differential equations are becoming more and more diverse. In recent years, nonlocal diffusion equations and a series of generalized nonlocal diffusion problems have attracted the interest and attention of many scientists. These nonlocal diffusion problems are widely used to describe the diffusion process, in which the density of a species at point x can be used to represent the probability distribution of moving from point y to point x. Convolution J ~ (U) ~ (X) T ~ (1) = ~ (1) RN ~ (J) ~ (x) -Y ~ (+) ~ = the velocity of species reaching point x In this paper, we study the properties of solutions of nonlocal diffusion equations with localized source terms, including the existence and uniqueness of solutions, the global existence of solutions and the finite time blow-up. The conditions for simultaneous and non-simultaneous blasting of the two components u and v of the solution and the estimation of the blow-up rate of the solution are discussed. In this paper, we first prove the existence and uniqueness of the solution by using the contraction mapping principle, and then establish a new comparison principle and derive the conditions for the solution to burst at finite time by using the method of upper and lower solutions. Then, by using some commonly used techniques of inequality and analysis and using the method in [1], we obtain the sufficient and necessary conditions for the simultaneous blow-up of T at finite time. Furthermore, the characterization of the blow-up mode and the set of blasting points for u and u are discussed, and finally, by means of comparison principle and ordinary differential equation inequality and other methods and techniques, In this paper, the sufficient conditions and necessary conditions for the simultaneous blasting of u and v at finite time T are derived, as well as the estimation of the blasting rate and the characterization of the set of blasting points.
【學(xué)位授予單位】：東南大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O175.29

【參考文獻】

相關(guān)期刊論文 前2條

1 ;Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms[J];Science in China,Ser.A;2005年02期

2 林支桂,謝春紅,王明新;THE BLOW-UP PROPERTIES OF SOLUTIONS TO A PARABOLIC SYSTEM WITH LOCALIZED NONLINEAR REACTIONS[J];Acta Mathematica Scientia;1998年04期

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本文編號：2048149