【摘要】：本文旨在研究幾類非線性發(fā)展方程的解的奇異性.首先研究了一類帶有奇異性的非線性拋物方程和一類帶有奇異性的半線性邊緣退化拋物方程的解的爆破性質(zhì),通過所構(gòu)建的一個介于能量泛函與Nehari泛函之間的函數(shù),得到了一個新的爆破條件,這個條件證明了當(dāng)初始能量大于臨界初始能量時方程的解爆破的可能性.然后研究了兩類帶有非局部項的p階雙調(diào)和方程,在適當(dāng)?shù)募僭O(shè)條件下,運用Galerkin逼近方法得到了弱解的全局存在性,然后利用一些關(guān)于非負(fù)函數(shù)的不等式得到了一些關(guān)于模型的弱解的爆破性、熄滅性和非熄滅性的結(jié)果.本文共分為五個章節(jié):第一章,主要介紹非線性發(fā)展方程的解的爆破性及熄滅性的研究概況及本文的研究目的、創(chuàng)新之處及方法.第二章,研究了一類帶有奇異性的非線性拋物方程的解的爆破性,不僅證明了方程的解在初始能量小于臨界初始能量的條件下是爆破的,而且證明了在初始能量不小于臨界初始能量的條件下方程的解的爆破性.第三章,研究了一類帶有奇異性的半線性邊緣退化拋物方程的弱解的爆破性.首先,介紹了邊緣型加權(quán)的p-Sobolev空間及一些引理.然后證明了方程的弱解在初始能量大于臨界初始能量的條件下是爆破的.第四章,研究了一類帶有非局部項|u|q-|Ω|-1∫Ω|u|qdx的p(p≤2)階雙調(diào)和方程的弱解的性質(zhì).本章不僅證明了方程在初始能量E(u0)≤0的條件下弱解的爆破性,而且在適當(dāng)?shù)某踔导僭O(shè)條件下證明了弱解的熄滅性與非熄滅性.第五章,研究了一類帶有非局部項|u|q-1u-|Ω|-1∫Ω|u|q-1udx的p(p2)階雙調(diào)和方程的弱解的性質(zhì).本章討論了方程的弱解的全局存在性及在初始能量E(u0)≤0和E(u0)0兩種情況下方程的弱解的爆破性,同時還考慮了方程的弱解的熄滅性與非熄滅性.最后,還對爆破時間的上界進(jìn)行了估計.
[Abstract]:The purpose of this paper is to study the singularity of solutions for some nonlinear evolution equations. In this paper, the blow-up properties of solutions for a class of nonlinear parabolic equations with singularities and a class of semilinear marginal degenerate parabolic equations with singularity are studied. A function between the energy functional and the Nehari functional is constructed. A new blasting condition is obtained, which proves the possibility of the solution blasting of the equation when the initial energy is greater than the critical initial energy. Then, two classes of p-order biharmonic equations with nonlocal terms are studied. Under appropriate assumptions, the global existence of weak solutions is obtained by using Galerkin approximation method. Then, by using some inequalities about nonnegative functions, we obtain some results on the blow-up, extinguishment and non-extinguishment of weak solutions of the model. This paper is divided into five chapters: the first chapter mainly introduces the study of the blow-up and extinguishment of the solution of the nonlinear evolution equation, the purpose, the innovation and the method of this paper. In chapter 2, we study the blow-up of solutions for a class of nonlinear parabolic equations with singularity. It is not only proved that the solution of the equation is blow-up under the condition that the initial energy is less than the critical initial energy. Moreover, it is proved that the solution of the equation is blow-up under the condition that the initial energy is not less than the critical initial energy. In chapter 3, we study the blow-up of weak solutions for a class of semilinear edge degenerate parabolic equations with singularity. Firstly, the edge weighted p-Sobolev space and some Lemma are introduced. Then it is proved that the weak solution of the equation is blasting under the condition that the initial energy is larger than the critical initial energy. In chapter 4, we study the weak solutions of a class of biharmonic equations of order p (p 鈮，

本文編號：2147383