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

Yuhua Lin

Studia Mathematica (1998)

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

Abstract

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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 t t = u x x + b ( t , x ) u x ( t , x ) + c ( t , x ) u ( t , x ) + f ( t , x ) f o r ( t , x ) ∈ [ 0 , T ] × [ 0 , 1 ] , u ( t , 0 ) = u ( t , 1 ) = 0 f o r t ∈ [ 0 , T ] , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = v 0 ( x ) f o r x ∈ [ 0 , 1 ] in the space of continuous functions on [0,1].

How to cite

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Lin, 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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  1. [1] G. Da Prato and E. Sinestrari, Differential operators with non dense domain, Ann. Scuola Norm. Sup. Pisa 14 (1987), 285-344. Zbl0652.34069
  2. [2] J. Kisyński, On cosine operator functions and one-parameter groups of operators, Studia Math. 44 (1972), 93-105. Zbl0232.47045
  3. [3] D. Lutz, On bounded time-dependent perturbation of operator cosine functions, Aequationes Math. 23 (1981), 197-203. Zbl0512.34047
  4. [4] I. Miyadera, S. Oharu and N. Okazawa, Generation theorems of semi-groups of linear operators, Publ. RIMS Kyoto Univ. 8 (1973), 509-555. Zbl0262.47030
  5. [5] H. Oka, Integrated resolvent operators, J. Integral Equations Appl. 7 (1995), 193-232. Zbl0846.45005
  6. [6] H. Serizawa and M. Watanabe, Perturbation for cosine families in Banach spaces, Houston J. Math. 12 (1986), 117-124. Zbl0607.47044
  7. [7] H. Serizawa and M. Watanabe, Time-dependent perturbation for cosine families in Banach spaces, ibid., 579-586. Zbl0619.47037
  8. [8] M. Sova, Cosine operator functions, Rozprawy Mat. 49 (1966). 
  9. [9] N. Tanaka, Quasilinear evolution equations with non-densely defined operators, Differential Integral Equations 9 (1996), 1067-1106. Zbl0942.34053

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