Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=−Au+ℱ(u)
Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=−Au+ℱ(u)
复制标题
DOI：
10.1007/bf00263041
复制
发表时间：
1973
影响因子：
2.5
通讯作者：
H. Levine
中科院分区：
文献类型：
--
作者：
H. Levine
许多作者[3]、[4]和[5]用例子证明了非线性抛物型方程对任意初值一般不具有整体解。施特劳斯[10]也指出，”有几个例子.也就是说，在大范围内不存在。然而，总的来说，可以作为理论指导的反例肯定是缺乏的。“本文的目的就是提供这样一个指南。我们将证明一些”抽象”定理，主要的一个大致说，如果u（.）是du-Au ~+，~（u（t）），t~[O，T），~（0）= 0，u（O）= uo（I. 1)其中P和A是定义在真实的或复Hilbert空间的稠密子域D上的”正”线性算子（见附录），其中o~满足以下抽象条件：
A number of authors [3],[4] and [5] have shown by example that, in general, nonlinear parabolic equations do not possess global solutions for arbitrary initial data. STRAUSS [10] has remarked also that" There are a few examples... of solutions which'blow up', that is, do not exist in the large. In general, however, there is a definite lack of counterexamples which can be used as guides to the theory." It is the purpose of this paper to give such a guide. We shall prove some" abstract" theorems, the principal one saying roughly that if u (.) is a strongly continuously differentiable solution of du-Au+,~(u (t)), t~[O, T),~(0)= 0, u (O)= uo (I. 1) P d~-= where P and A are" positive" linear operators defined on a dense subdomain D of a real or complex Hilbert space (see the Appendix), where o~ satisfies the following abstract condition: