Existence and Representation Theorems for a Semilinear Sobolev Equation in Banach Space
Existence and Representation Theorems for a Semilinear Sobolev Equation in Banach Space
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DOI：
10.1137/0503051
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发表时间：
1972-08
影响因子：
2
通讯作者：
R. Showalter
中科院分区：
文献类型：
--
作者：
R. Showalter
利用Sobolev型偏微分方程的边值问题，建立了Banach空间中半线性发展方程解的存在性定理。算子被假定为可测的，并满足强制性估计，这不一定是一致的，在其时间依赖性，并满足Lipschitz条件的非线性项。应用程序简要说明。1.导论.在可分自反Banach空间中考虑非线性发展方程f/（t）u '（t）+（t）u（t）f（t，u（t））的抽象Cauchy问题.假设线性算子f（t）在Banach空间中是弱可测的，并且满足Banach空间上的非一致强制估计，这允许它们对某些值退化.假设线性算子族&（t）在t中是弱可测的.非线性项f（t，u）在u中是可测的，Lipschitz在u中是可测的.有三种类型的解被认为是弱解、温和解和强解。温和解（本质上）是一个弱解，它允许某种积分表示，我们将证明这两个概念通过可测性假设而不同。一个强解是一个弱解，对于这个弱解，方程中的每一项几乎对每一个t都属于一个指定的希尔伯特空间。本文的研究计划如下。第2节包含我们将使用的一些技术结果和符号。其中包括向量和算子值函数的可测性、Gronwall不等式和Banach空间值函数的一个初等不动点定理。文[3]定义了弱解的概念，在各种假设下得到了弱解的唯一性、局部存在性和整体存在性的结果。这些结果在4中被用来构造线性传播子（它解决了线性方程与f 0），从而引入了温和的解决方案的概念。我们证明了温和的解决方案（本地和全球）存在相同的假设用于弱解的存在。强解在5中引入。我们给出了弱解是强解的充分条件;这些条件本质上是算子/（t）支配算子&（t）。最后，我们独立地得到了强解存在（和唯一）的一个充分条件;这个条件要求函数f被算子f（t）支配。
An existence theory is developed for a semilinear evolution equation in Banach space which is modeled on boundary value problems for partial differential equations of Sobolev type. The operators are assumed to be measurable and to satisfy coercive estimates which are not necessarily uniform in their time dependence, and to satisfy Lipschitz conditions on the nonlinear term. Applications are briefly indicated. 1. Introduction. We shall consider the abstract Cauchy problem for the nonlinear evolution equation ////(t)u'(t) + (t)u(t) f(t, u(t)) in a separable and reflexive Banach space. The linear operators /(t) are assumed to be weakly measurable in and to satisfy nonuniform coercive estimates over the Banach space which permit them to degenerate for certain values of t. The family of linear operators &(t) are assumed to be weakly measurable in t. The nonlinear term f(t, u) is measurable in and Lipschitz in u. Three types ofsolution are considered weak, mild, and strong.A mild solution is (essentially) a weak solution which permits a certain integral representation, and we shall prove that these two notions differ by a measurability assumption. A strong solution is a weak solution for which each term in the equation belongs to a specified Hilbert space for almost every t. The plan of the paper is as follows. Section 2 contains some technical results and notation we shall use. These include measurability of vector- and operator- valued functions, Gronwall's inequality, and an elementary fixed-point theorem for Banach space-valued functions. The weak solution is defined in 3, where we obtain results on uniqueness, local existence and global existence under various hypotheses. These results are used in 4 to construct the linear propagator (which resolves the linear equation withf 0) and thereby to introduce the notion of a mild solution. We prove that mild solutions (local and global) exist with the same hypotheses as used for existence of weak solutions. Strong solutions are introduced in 5. We give sufficient conditions for a mild solution to be strong; these conditions are essentially that the operators ///(t) dominate the operators &(t). Finally we obtain independently a sufficient condition for the existence (and uniqueness) of a strong solution; this condition requires that the function f be dominated by the operators /(t).