摘要
The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.
讨论Banach空间X上二阶抽象微分方程d^2/（dr^2）u（t，x）=Au（t，x）；u（0，x）=x，d/（dt）u（0，x）=0，x∈X的不适定情况，这里A是X上的闭算子；引进空间Y（A，k），即使得二阶抽象微分方程有次弱解v（t,x），且满足ess sup{（1＋t）^-k｜d/（dt）〈v（t,x），x^＊〉｜：t≥0，x^＊∈X^＊，｜x^＊‖≤1}〈＋∞的x∈X的全体，及空间H（A，ω），即使得二阶抽象微分方程有次弱解v（t，x），且满足的x∈X的全体．证明了如下结论：Y（A，k）和H（A，ω）均为Banach空间，且Y（A，k）和H（A，ω）均连续嵌入X；A在Y（A，k）上的限制算子A｜Y（A，k）生成一个一次积分Cosine算子函数{（t））t≥0，满足limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k，任意t≥0；A在H（A，ω）上的限制算子A｜H（A,ω）生成一个一次积分Cosine算子函数{C（t）}t≥0，满足limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A，ω）≤≤Me^ωt，任意t≥0.
基金
The Natural Science Foundation of Department ofEducation of Jiangsu Province (No06KJD110087)
