【摘要】 受文de Laubenfels[1](1997,Isreal Journal ofM athem atics,98:189~207)的启发,引进空间W(A,k)和H(A,ω),它们分别是使得该二阶抽象Cauchy问题有在[0,∞)一致连续且O((1+t)k)有界和O(eωt)有界的弱解的x∈X的全体.讨论Banach空间X上二阶抽象Cauchy问题的具有多项式有界解或指数有界解的极大子空间问题.由W ang and W ang[2](1996,Functional Analysis in Ch ina.K luwer,333~350)知,该Cauchy问题适定的充要条件是该Cauchy问题中的X上闭算子A生成一个强连续Cosine算子函数.处理该Cauchy问题不适定的情况.证明或指出了如下结论:.W(A,k)和H(A,ω)均为Banach空间,且W(A,k)和H(A,ω)均连续嵌入X;.部分算子A|W(A,k)生成一个多项式有界的余弦算子函数{C(t)}t∈R+,使‖C(t)‖W(A,k)≤2(1+t)k;.部分算子A|H(A,ω)生成一个指数有界的余弦算子函数{C(t)}t∈R+,使‖C(t)‖H(A,ω)≤2eωt;.W(A,k)和H(A,ω)分别是极大的.即若有Banach空间Y连续嵌入X,且使A|Y生成一个O((1+t)k)有界的余弦算子函数,那么Y连续嵌入W(A,k);而若使A|Y生成一个O(eωt)有界的余弦算子函数,那么Y连续嵌入H(A,ω).更多还原

【Abstract】 This paper is devoted to discuss the topic on maximal subspaces for the polynomially or exponentially bounded mild solutions of the following abstract Cauchy problem,d²[]dt²u（t,x）=Au（t,x） u（0,x）=x,u’（0,x）=0（*）where A is a closed operator on a Banach space X.It follows from Wang and Wang^[2]（1996,Functional Analysis in China.Kluwer,333～350） that the Cauchy problem（*） is well-posed,if and only if the closed operator occurring in（*）,A,generates a strongly continuous Cosine operator function.In this paper we treat the case that（*） is ill-posed.Motivated by de Laubenfels^[1]（1997,Isreal Journal of Mathematics,98:189～207）,we introduce two subspaces W（A,k） and H（A,ω）.W（A,k） is the set of all x in X for which the equation（*） has a mild solution u（t,x） such that（1+t）^-ku（t,x） is uniformly continuous and bounded on [0,∞).And H（A,ω） is the set of all x in X for which the equation（*） has a mild solution u（t,x） such that e^{-ω t}u（t,x） is uniformly continuous and bounded on [0,∞).We prove or point out the following conclusions.· W（A,k） and H（A,ω） are Banach spaces,and both are continuously embedded in X;· The part operator A|_W（A,k） generates a polynomially bounded Cosine operator function {C（t）}_t∈R⁺ such that‖C（t）‖_W（A,k）≤ 2（1+t）^k; · The part operator A|_H（A,ω） generates an exponently bounded Cosine operator function {C（t）}_t∈R⁺ such that‖C（t）‖_H（A,ω）≤ 2e^{ω t}; · The subspaces of X,W（A,k） and H（A,ω） are respectively maximal in the sense that,let Y be another subspace continuously embedded in X,if A|_Y generates an O（（1+t）^k） bounded Cosine operator function then Y is continuously embedded in W（A,k）,or if A|_Y generates an O(e^{ω t}) bounded Cosine operator function then Y is continuously embedded in H（A,ω）.更多还原