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ISSN 1088-6850(e) ISSN 0002-9947(p)

Approximation theorems for the propagators of higher order abstract Cauchy problems

Author(s): Jin Liang; Rainer Nagel; Ti-Jun Xiao
Journal: Trans. Amer. Math. Soc. 360 (2008), 1723-1739.
MSC (2000): Primary 34G10; Secondary 35R20, 47D09
Posted: November 26, 2007
MathSciNet review: 2366960
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Abstract: In this paper, we present two quite general approximation theorems for the propagators of higher order (in time) abstract Cauchy problems, which extend largely the classical Trotter-Kato type approximation theorems for strongly continuous operator semigroups and cosine operator functions. Then, we apply the approximation theorems to deal with the second order dynamical boundary value problems.

References:

1.: K. T. Andrews, K. L. Kuttler, and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl. 197 (1996), 781-795. MR 1373080 (96m:34116)
2.: W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs Math. 96, Birkhäuser Verlag, 2001. MR 1886588 (2003g:47072)
3.: H. T. Banks and D. J. Inman, On damping mechanisms in beams, ASME Trans. 58 (1991), 716-723.
4.: A. Batkai and K.-J. Engel, Abstract wave equations with generalized Wentzell boundary conditions, J. Differential Equations 207 (2004), 1-20. MR 2100812 (2005g:34124)
5.: R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, New York, 1976. MR 0460842 (57:834)
6.: V. Casarino, K.-J. Engel, R. Nagel, and G. Nickel, A semigroup approach to boundary feedback systems, Integral Equations Operator Theory 47 (2003), 289-306. MR 2012840 (2004j:34128)
7.: K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, GTM 194, Springer-Verlag, Berlin, New York, 2000. MR 1721989 (2000i:47075)
8.: J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Part. Diff. Equations 18 (1993), 1309-1364. MR 1233197 (94g:35112)
9.: H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, Elsevier Science Publishers B. V., Amsterdam, 1985. MR 797071 (87b:34001)
10.: A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, $C\sb 0$ -semigroups generated by second order differential operators with general Wentzell boundary conditions, Proc. Amer. Math. Soc. 128 (2000), 1981-1989. MR 1695147 (2000m:47054)
11.: A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Generalized Wentzell boundary conditions and analytic semigroups in , In: Semigroups of Operators: Theory and Applications (Newport Beach, CA, 1998), 125-131, Progr. Nonlinear Diff. Equations Appl. 42, Birkhäuser, Basel, 2000. MR 1788874 (2001h:47061)
12.: A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evolution Equations 2 (2002), 1-19. MR 1890879 (2003b:35089)
13.: A. Favini and E. Obrecht, Conditions for parabolicity of second order abstract differential equations, Diff. Integral Equations 4 (1991), 1005-1022. MR 1123349 (92m:47078)
14.: C. L. Frota and J. A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions, J. Differential Equations 164 (2000), 92-109. MR 1761418 (2001h:35126)
15.: J. A. Goldstein, On the convergence and approximation of cosine functions, Aequationes Math. 10 (1974), 201-205. MR 0358435 (50:10901)
16.: J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Math. Monographs, Oxford Univ. Press, New York, 1985. MR 790497 (87c:47056)
17.: C. G. Gal, G. R. Goldstein, and J. A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, J. Evolution Equations 3 (2003), 623-635. MR 2058054 (2005g:35188)
18.: K. Ito and F. Kappel,, The Trotter-Kato theorem and approximation of PDEs, Math. Comp. 67 (1998), 21-44. MR 1443120 (98e:47060)
19.: T. Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad. 35 (1959), 467-468. MR 0117570 (22:8347)
20.: T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
21.: M. Kramar, D. Mugnolo, and R. Nagel, Theory and applications of one-sided coupled operator matrices, Conf. Semin. Mat. Univ. Bari 283 (2002), 1-29. MR 1966540 (2004e:47062)
22.: T. Kurtz, Extensions of Trotter's operator semigroup approximation theorems, J. Funct. Anal. 3 (1969), 111-132. MR 0242016 (39:3351)
23.: T. Kurtz, A general theorem on the convergence of operator semigroups, Trans. Amer. Math. Soc. 148 (1970), 201-205. MR 0256210 (41:867)
24.: J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50 (1983), 163-182. MR 719445 (85f:35025)
25.: J. Liang, R. Nagel and T. J. Xiao, Nonautonomous heat equations with generalized Wentzell boundary conditions. J. Evolution Equations 3 (2003), 321-331. MR 1980980 (2004b:35134)
26.: J. L. Lions, Equations différentielles opérationnelles et problèmes aux limites, Springer-Verlag, Berlin, 1961. MR 0153974 (27:3935)
27.: C. Lizama, On an extension of the Trotter-Kato theorem for resolvent families of operators, J. Integral Equations Appl. 2 (1990), 269-280. MR 1045773 (91c:47093)
28.: C. Lizama, On approximation and representation of -regularized resolvent families, Integral Equations Operator Theory 41 (2001), 223-229. MR 1847173 (2002f:47085)
29.: S. Nicaise, The Hille-Yosida and Trotter-Kato theorems for integrated semigroups, J. Math. Anal. Appl. 180 (1993), 201-205. MR 1251861 (94m:47082)
30.: A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. MR 710486 (85g:47061)
31.: S. Piskarev and V. V. Vasiliev, Differential equations in Banach spaces. II. Theory of cosine operator functions, Functional analysis. J. Math. Sci. (N. Y.) 122 (2004), 3055-3174. MR 2084186 (2005d:34133)
32.: T. I. Seidman, Approximation of operator semi-groups, J. Funct. Anal. 5 (1970), 160-166. MR 0254659 (40:7866)
33.: H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887-919. MR 0103420 (21:2190)
34.: T. J. Xiao and J. Liang, The Cauchy Problem for Higher Order Abstract Differential Equations, Lect. Notes in Math., vol. 1701, Springer, Berlin, New York, 1998. MR 1725643 (2001a:34099)
35.: T. J. Xiao and J. Liang, Approximations of Laplace transforms and integrated semigroups, J. Funct. Anal. 172 (2000), 202-220. MR 1749872 (2001b:47072)
36.: T. J. Xiao and J. Liang, A solution to an open problem for wave equations with generalized Wentzell boundary conditions, Math. Ann. 327 (2003), 351-363. MR 2015075 (2004m:35162)
37.: T. J. Xiao and J. Liang, Complete second order differential equations in Banach spaces with dynamic boundary conditions, J. Differential Equations 200 (2004), 105-136. MR 2046319 (2005f:34167)
38.: T. J. Xiao and J. Liang, Second order parabolic equations in Banach spaces with dynamic boundary conditions, Trans. Amer. Math. Soc. 356 (2004), 4787-4809. MR 2084398 (2005e:34173)
39.: K. Yosida, Functional Analysis (6th edition), Springer-Verlag, New York, 1980. MR 617913 (82i:46002)

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