几类非线性发展方程解的奇异性质的研究
发布时间：2018-07-27 09:32
【摘要】：本文旨在研究几类非线性发展方程的解的奇异性.首先研究了一类带有奇异性的非线性抛物方程和一类带有奇异性的半线性边缘退化抛物方程的解的爆破性质,通过所构建的一个介于能量泛函与Nehari泛函之间的函数,得到了一个新的爆破条件,这个条件证明了当初始能量大于临界初始能量时方程的解爆破的可能性.然后研究了两类带有非局部项的p阶双调和方程,在适当的假设条件下,运用Galerkin逼近方法得到了弱解的全局存在性,然后利用一些关于非负函数的不等式得到了一些关于模型的弱解的爆破性、熄灭性和非熄灭性的结果.本文共分为五个章节:第一章,主要介绍非线性发展方程的解的爆破性及熄灭性的研究概况及本文的研究目的、创新之处及方法.第二章,研究了一类带有奇异性的非线性抛物方程的解的爆破性,不仅证明了方程的解在初始能量小于临界初始能量的条件下是爆破的,而且证明了在初始能量不小于临界初始能量的条件下方程的解的爆破性.第三章,研究了一类带有奇异性的半线性边缘退化抛物方程的弱解的爆破性.首先,介绍了边缘型加权的p-Sobolev空间及一些引理.然后证明了方程的弱解在初始能量大于临界初始能量的条件下是爆破的.第四章,研究了一类带有非局部项|u|q-|Ω|-1∫Ω|u|qdx的p(p≤2)阶双调和方程的弱解的性质.本章不仅证明了方程在初始能量E(u0)≤0的条件下弱解的爆破性,而且在适当的初值假设条件下证明了弱解的熄灭性与非熄灭性.第五章,研究了一类带有非局部项|u|q-1u-|Ω|-1∫Ω|u|q-1udx的p(p2)阶双调和方程的弱解的性质.本章讨论了方程的弱解的全局存在性及在初始能量E(u0)≤0和E(u0)0两种情况下方程的弱解的爆破性,同时还考虑了方程的弱解的熄灭性与非熄灭性.最后,还对爆破时间的上界进行了估计.
[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，
本文编号：2147383
[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，
本文编号：2147383
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