摘要
介绍了一类椭圆型方程的非局部特征值问题 .通过在一个Sobolev空间选取特殊基 ,并利用Faedo Galerkin方法和能量估计讨论了问题当F(u ,x ,t)是一类具体函数时的整体解的存在性与唯一性 ;当F(u ,x ,t) =｜u｜p-1u时 ,研究了解的爆破问题 ;最后 ,利用极大单调算子理论证明了问题的关于时间的周期解的存在性与唯一性 .
A non local eigenvalue problem is studied for a kind of elliptic equations. Since the maximum principle in partial differential equations is invalid for this kind of equations, the super sub solution method can not be applied. By choosing a special base in a Sobolev space, applying the Faedo Galerkin method and the energy estimates we get, in Section 3, the existence and the uniqueness of the global solution to the above problem for certain functions F. In Section 4, we consider the blowing up situation of the solution to the problem for a specified F. We get that if the initial data u 0≠0 and the energy of u is non positive, then the solution of the problem must blow up in a finite time. Finally, using the theory of maximal monotonic operators, the existence and uniqueness of the periodic solution to the problem is considered and it is proved that there is only one periodic solution provided f(x,t)∈L p′ (Q) where Q=Ω×(0, T).
基金
国家自然科学基金!资助项目 (10 0 710 80 )
