A semilinear Sobolev evolution equation in a Banach space
A semilinear Sobolev evolution equation in a Banach space
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DOI：
10.1016/0022-0396(77)90009-2
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发表时间：
1977-06
影响因子：
2.4
通讯作者：
H. Brill
中科院分区：
文献类型：
--
作者：
H. Brill
[~~W4+au(T)=g(@)，q，(1.1)其中B和A是线性算子，其中域包含在实Banach空间X中，值域包含在实Banach空间Y中。G映射X[0，2‘1，T>；0到Y。本文的主要目的是在算子B支配算子A：D(B)CD(A)的基本条件下，发展了(1.1)的存在性理论。第三节利用Schauder不动点原理得到了连续函数g：xx[0，T]+y(1.1)的局部解，并证明了满足线性增长条件的非线性项g的整体解的存在性。我们在这里考虑的抽象问题是伪抛物型偏微分方程初边值问题的实现。
[~~ W4+ Au (t)= g (@), q,(1.1) in which B and A are linear operators with domains contained in a real Banach space X and ranges contained in a real Banach space Y. g maps X x[0, 2’1, T> 0, into Y. The main purpose of this note is to develop an existence theory for (1.1) under the fundamental requirement that the operator B dominate the operator A: D (B) CD (A).In the second section the additional assumptions on the operators B and A are collected and some a priori estimates are proved. Schauder’s fixed point principle is used in Section 3 in order to obtain a local solution of (1.1) for each continuous function g: X x[0, T]+ Y. Moreover, we establish the existence of a global solution for nonlinearities g satisfying a linear growth condition. The abstract problem we consider here arises as a realization of an initial boundary value problem for the pseudoparabolic partial differential equation