[1] Wang Meiying,. Subspaces for weak mild solutionsof the second order abstract differential equation [J]. Journal of Southeast University (English Edition), 2007, 23 (2): 313-316. [doi:10.3969/j.issn.1003-7985.2007.02.032]

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Subspaces for weak mild solutionsof the second order abstract differential equation()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

23

Issue:

2007 2

Page:

313-316

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2007-06-30

Info

Title:

Subspaces for weak mild solutionsof the second order abstract differential equation

Author(s):

Wang Meiying

Department of Applied Mathematics, Nanjing Audit University, Nanjing 210029, China

Keywords:

second order abstract differential equation;  polynomially bounded solution;  cosine operator function

PACS:

O177.5

DOI:

10.3969/j.issn.1003-7985.2007.02.032

Abstract:

The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem (d^2)/(2dt^2)u2(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 supe^-ωt|(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≥00 such that lim^—_h→0⁺(1)/h=C(t+h)-C(t)=_{Y(A, k)≤M}(, )(1+t)^k, ∠t≥0; the restriction operator A_{H(A, ω)} (, )generates a once-integrated cosine operator family {C(t)}_t≥0 0such that lim^—_h→0⁺1/h=C(t+h)-C(t)=_{H(A, ω)≤M}(, )e^ωt, ∠t≥0.

References:

[1] de Laubenfels R.Existence families, functional calculi and evolution equations[C]//Lecture Notes in Mathematics.Berlin:Springer-Verlag, 1994, 1570:27-54.
[2] Arendt W, Batty Ch J K, Hieber M, et al.Vector-valued Laplace transforms and Cauchy problems[M].Birkhauser, 2001.
[3] de Laubenfels R, Huang Z, Wang S, et al.Laplace transforms of polynomially bounded vector-valued functions and semigroups of operators [J].Israel Journal of Mathematics, 1997, 98:189-207.
[4] Wang Shenwang.Hille-Yosida type theorems for local regularized semigroups and local integrated semigroups[J].Studia Mathematica, 2002, 152(1):45-67.
[5] Davies E B, Pang M M.The Cauchy problem and a generalization of the Hille-Yosida theorem[J].Proc London Math Soc, 1987, 55:181-208.
[6] Arendt W.Vector-valued Laplace transforms and Cauchy problems [J].Israel Journal of Mathematics, 1987, 59:327-352.
[7] Wang Meiying, Jiang Huikun.Maximal subspaces for polynomially bounded solutions of the second order abstract differential equation[J].Journal of Nanjing University:Natural Science Edition, 2006, 42(1):11-16.(in Chinese)
[8] Wang Shenwang, Wang Meiying, Shen Yan.Perturbation theorems for local integrated semigroups and their applications[J].Studia Mathematica, 2005, 170(2):121-146.

Memo

Memo:

Biography: Wang Meiying(1950—), female, professor, nsjwangmy@sina.com.

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