中文摘要：

受文de Laubenfels（1997，Isreal Journal of Mathematics，98：189—207）的启发，引进空间形（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（e^ωt）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Wang and Wang（1996，Functional Analysis in China．Kluwer，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，∞）均连续嵌入X； ·部分算子AIW（A，k）生成一个多项式有界的余弦算子函数使‖C（t）‖W（A,k）≤2（1＋t）^k；·部分算子AIW（A，ω）生成一个指数有界的余弦算子函数{C（t）}t∈R＋，‖C（t）‖H（W,ω）≤2e^ωt；·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AIY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AIY生成一个O（e^ωt）有界的余弦算子函数，那么Y连续嵌入H（A，ω）．

英文摘要：

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,{d^2/dt^2u（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 （ 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 （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）^-k u（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^-ωt 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│W（A,ω） generates an exponcntly bounded Cosine operator function {C（t）}t∈R＋ such that ‖C（t）‖H（A,ω）≤2e^ωt;·The subspaces of X, W（A,k） and ll（A,ω） are respectively maximal in the sense that, let Y be another subspace continuously embedded in X, if A I r generates an O （ （ 1 ＋ t） ^k ） bounded Cosine operator function then Y is continuously embedded in W（A ,k）, or if Air generates an O（e^ωt） bounded Cosine operator function then Y is continuously embedded in H（ A,ω）.