# zbMATH — the first resource for mathematics

Integrated semigroups and their application to complete second order Cauchy problems. (English) Zbl 0686.47038
The author studies well posedness of the complete second order Cauchy problem $u''(t)-Bu'(t)-Au(t)=0;\quad u(0)=x,\quad u'(0)=y.$ The main concept of the paper is “biclosedness” of the pair A, B. That means $$P(\lambda)=\lambda^ 2I-\lambda B-A$$ with domain $$D(P)(\lambda)=D(A)\cap D(B)$$ is closed for all $$\lambda >w$$. Of course estimations concerning the resolvent $$R_{\lambda}=P^{-1}(\lambda)$$ also play a crucial role.
Reviewer: J.de Graaf

##### MSC:
 47D03 Groups and semigroups of linear operators 47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX) 34G10 Linear differential equations in abstract spaces
Full Text:
##### References:
 [1] W. Arendt,Vector valued Lrplace transforms and Cauchy problems, Israel J. Math.59 (1987), 327–352. · Zbl 0637.44001 · doi:10.1007/BF02774144 [2] W. Arendt, H. Kellermann,Integrated solutions of Volterra integro-differential equations and applications, Preprint 1987. · Zbl 0675.45017 [3] P. Aviles, J. Sandefur,Nonlinear second order equations with applications to partial differential equations, J. Diff. Eqns.58 (1985), 404–427. · Zbl 0572.34004 · doi:10.1016/0022-0396(85)90008-7 [4] R. Beals,On the abstract Cauchy problem, J. Func. Anal.10 (1972), 281–299. · Zbl 0239.34028 · doi:10.1016/0022-1236(72)90027-4 [5] P. Brenner,The Cauchy problem for symmetric hyperbolic systems in L p Math. Scand.19 (1966), 27–37. · Zbl 0154.11304 [6] G. Chen, D.L. Russell,A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. (1982), 433–454. · Zbl 0515.73033 [7] I. Ciorąnescu,On the abstract Cauchy problem for the operator d 2 /dt 2 , Integral Equations Operator Theory7 (1984), 27–35. · Zbl 0532.47028 · doi:10.1007/BF01204911 [8] G. Da Prato,Semigruppi regolarizzabili, Ricerche Mat.15 (1966), 223–246. [9] E.B. Davies and M.M. Pang,The Cauchy problem and a generalization of the Hille-Yosida Theorem, Proc. London Math. Soc.55 (1987), 181–208. · Zbl 0651.47026 · doi:10.1112/plms/s3-55.1.181 [10] R. deLaubenfels,The relationship between C-semigroups and integrated semigroups, Preprint 1988. [11] R. deLaubenfels,C-semigroups and the Cauchy problem, Preprint 1988. [12] K. Engel,Polynomial operator matrices, Dissertation, Univ. Tübingen 1988. · Zbl 0686.47019 [13] H. Engler, F. Neubrander, J. Sandefur,Second order strongly damped quasilinear equations, In: T.L. Gill, W.W. Zachary (eds.). ”Nonlinear Semigroups, Partial Differential Equations and Attractors,” Lecture Notes1248, Springer-Verlag, 1987. · Zbl 0614.34055 [14] H. Falun, L. Kangsheng,On the mathematical model for linear elastic systems with structural damping, Preprint 1987. · Zbl 0659.34054 [15] H.O. Fattorini, ”The Cauchy Problem,” Addison Wesley, Reading, Mass., 1983. · Zbl 0493.34005 [16] H.O. Fattorini, ”Second Order Linear Differential Equations in Banach Spaces,” North-Holland Mathematics Studies108, 1985. · Zbl 0564.34063 [17] J.A. Goldstein, ”Semigroups of Linear Operators and Applications,” Oxford University Press, New York, 1985. · Zbl 0592.47034 [18] R. Hersh,Explicit solutions of a class of higher order abstract Cauchy problems, J. Diff. Eqns.8 (1970), 570–579. · Zbl 0208.38603 · doi:10.1016/0022-0396(70)90030-6 [19] M. Hieber, Dissertation, Univ. Tübingen. To appear. [20] H. Kellermann,Integrated semigroups, Dissertation, Univ. Tübingen, 1986. · Zbl 0604.47025 [21] J. Kisynski,On cosine operator functions and one-parameter groups of operators, Studia Math.44 (1972), 93–105. · Zbl 0232.47045 [22] J.L. Lions,Les semi-groupes distributions, Portugaliae Mathematica19 (1960), 141–164. · Zbl 0103.09001 [23] W. Littman,The wave operator and L p norms, J. Math. Mech.12 (1963), 55–68. · Zbl 0127.31705 [24] I. Miyadera,On the generators of exponentially bounded C-semigroups, Proc. Japan Acad.62 (1986), 239–242. · Zbl 0617.47032 [25] I. Miyadera, S. Oharu, and N. Okazawa,Generation theorems of semigroups of linear operators, Publ. Res. Inst. Math. Sci., Kyoto Univ.8 (1973), 509–555. · Zbl 0262.47030 · doi:10.2977/prims/1195192960 [26] I.V. Mel’nikova, A.I. Filinkoy,Classification and well-posedness of the Cauchy problem for second-order equations in a Banach space, Soviet Math. Dokl.29 (1984), 646–651. [27] R. Nagel,Towards a ”matrix theory” for unbounded operator matries, Math. Z. To appear. [28] F. Neubrander,Wellposedness of higher order abstract Cauchy problems, Trans. Amer. Math. Soc.295 (1986), 257–290. · Zbl 0589.34004 · doi:10.1090/S0002-9947-1986-0831199-8 [29] F. Neubrander,Integrated semigroups and their applications to the abstract Cauchy problem, Pacific J. Math. To appear. · Zbl 0675.47030 [30] E. Obrecht,Sul problema di Cauchy per le equazioni paraboliche astratte di ordine n, Rend. Sem. Mat. Univ. Padova,53 (1975), 231–256. · Zbl 0326.34076 [31] J. Sandefur,Existence and Uniqueness of solutions of second order nonlinear differential equations, SIAM J. Math. Anal.14 (1983), 477–487. · Zbl 0513.34069 · doi:10.1137/0514041 [32] M. Sova,Probleme de Cauchy pour equations hyperboliques operation-nelles a coefficients constants non-bornes, Ann. Scuola Norm. Suo. Pisa,22 (1968), 67–100. · Zbl 0202.10902 [33] M. Sova,Problemes de Cauchy paraboliques abstraits de classes superieures et les semi-groupes distributions, Ricerche Mat.18 (1969), 215–238. · Zbl 0196.16301 [34] T. Takenaka, N. Okazawa,Abstract Cauchy problems for second order linear differential equations in a Banach space, Hiroshima Math. J.17 (1987), 591–612. · Zbl 0645.47041 [35] N. Tanaka,On exponentially bounded C-semigroups, Tokyo J. Math.10 (1987), 107–117. · Zbl 0631.47029 · doi:10.3836/tjm/1270141795 [36] N. Tanaka, I. Miyadera,Some remarks on C-semigroups and integrated semigroups, Proc. Japan Acad.63 (1987), 139–142. · Zbl 0642.47034 · doi:10.2183/pjab.63.348 [37] H.R. Thieme,Integrated semigroups and duality, Preprint 1987. [38] C.C. Travis, G.F. Webb,Cosine families and abstract nonlinear second order differential equations, Acta Math. Sci. Hung.32 (1978), 75–96. · Zbl 0388.34039 · doi:10.1007/BF01902205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.