# Time-dependent perturbation theory for abstract evolution equations of second order

Studia Mathematica (1998)

- Volume: 130, Issue: 3, page 263-274
- ISSN: 0039-3223

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topLin, Yuhua. "Time-dependent perturbation theory for abstract evolution equations of second order." Studia Mathematica 130.3 (1998): 263-274. <http://eudml.org/doc/216557>.

@article{Lin1998,

abstract = {A condition on a family $\{B(t):t ∈ [0,T]\}$ of linear operators is given under which the inhomogeneous Cauchy problem for u"(t)=(A+ B(t))u(t) + f(t) for t ∈ [0,T] has a unique solution, where A is a linear operator satisfying the conditions characterizing infinitesimal generators of cosine families except the density of their domains. The result obtained is applied to the partial differential equation \[ u\_\{tt\} = u\_\{xx\} + b(t,x)u\_x(t,x) + c(t,x)u(t,x) + f(t,x) for (t,x) ∈ [0,T]×[0,1], u(t,0) = u(t,1) = 0 for t ∈ [0,T], u(0,x) = u\_0(x), u\_t(0,x) = v\_0(x) for x ∈ [0,1] \]
in the space of continuous functions on [0,1].},

author = {Lin, Yuhua},

journal = {Studia Mathematica},

keywords = {inhomogeneous Cauchy problem; infinitesimal generators of cosine families; partial differential equation},

language = {eng},

number = {3},

pages = {263-274},

title = {Time-dependent perturbation theory for abstract evolution equations of second order},

url = {http://eudml.org/doc/216557},

volume = {130},

year = {1998},

}

TY - JOUR

AU - Lin, Yuhua

TI - Time-dependent perturbation theory for abstract evolution equations of second order

JO - Studia Mathematica

PY - 1998

VL - 130

IS - 3

SP - 263

EP - 274

AB - A condition on a family ${B(t):t ∈ [0,T]}$ of linear operators is given under which the inhomogeneous Cauchy problem for u"(t)=(A+ B(t))u(t) + f(t) for t ∈ [0,T] has a unique solution, where A is a linear operator satisfying the conditions characterizing infinitesimal generators of cosine families except the density of their domains. The result obtained is applied to the partial differential equation \[ u_{tt} = u_{xx} + b(t,x)u_x(t,x) + c(t,x)u(t,x) + f(t,x) for (t,x) ∈ [0,T]×[0,1], u(t,0) = u(t,1) = 0 for t ∈ [0,T], u(0,x) = u_0(x), u_t(0,x) = v_0(x) for x ∈ [0,1] \]
in the space of continuous functions on [0,1].

LA - eng

KW - inhomogeneous Cauchy problem; infinitesimal generators of cosine families; partial differential equation

UR - http://eudml.org/doc/216557

ER -

## References

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- [7] H. Serizawa and M. Watanabe, Time-dependent perturbation for cosine families in Banach spaces, ibid., 579-586. Zbl0619.47037
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- [9] N. Tanaka, Quasilinear evolution equations with non-densely defined operators, Differential Integral Equations 9 (1996), 1067-1106. Zbl0942.34053

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