具有非线性边界耗散的四阶方程整体解的不存在性分析

发布时间：2018-06-13 22:04

本文选题：物理学 + 非线性耗散方程　；参考：《科技通报》2017年09期

【摘要】：物理学新现象的出现,在该领域内产生了对孤立子及混沌问题的探究,并出现了一些带有非线性耗散的方程,因此文中将具有非线性边界耗散的四阶方程作为研究对象,并对该方程的整体解进行不存在性分析。首先,运用变分法获取整体弱解的存在性,将Gronwall不等式与Galerkin方法和积分估计法结合进行恰当的先验预估计并研究解的渐近特性,通过积分不等式利用Sobolev嵌入定理和吸引子存在定理证明在内积空间中整体吸引子的存在性,同时得到了吸引子的存在条件;其次,引进位势井和井外集合,运用H?lder不等式与Galerkin方法结合给定初始能量条件,得到整体解存在的门槛结果,在该方程及给定的初始条件满足区间内单调递增条件时,利用反证法可证明方程解不存在整体解,即局部解可在限定时间内实现爆破。
[Abstract]:With the emergence of new phenomena in physics, the soliton and chaos problems have been explored in this field, and some equations with nonlinear dissipation have appeared. Therefore, the fourth order equation with nonlinear boundary dissipation is regarded as the research object in this paper. The nonexistence of the global solution of the equation is analyzed. Firstly, the existence of the global weak solution is obtained by using the variational method, the Gronwall inequality is combined with the Galerkin method and the integral estimation method to obtain the proper prior pre-estimate and the asymptotic properties of the solution are studied. By using Sobolev embedding theorem and attractor existence theorem, the existence of global attractor in inner product space is proved, and the existence condition of attractor is obtained. The threshold result of the existence of the global solution is obtained by combining the Hillder inequality with the Galerkin method with the given initial energy condition. When the equation and the given initial conditions satisfy the monotone increasing condition in the interval, the non-existence of the global solution can be proved by using the counter-proof method. That is, the local solution can realize the blasting in the limited time.
【作者单位】：南昌理工学院公共教学部;
【分类号】：O175

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