Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Compactness of the solution operator for a linear evolution equation with distributed measures

Author: Ioan I. Vrabie
Journal: Trans. Amer. Math. Soc. 354 (2002), 3181-3205
MSC (2000): Primary 47D06, 46G10, 47B07; Secondary 35A05, 35J99, 35K99
Published electronically: April 1, 2002
MathSciNet review: 1897396
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main goal of the present paper is to define the solution operator $(\xi ,g)\mapsto u$ associated to the evolution equation $du=(Au)dt+dg$, $u(0)=\xi$, where $A$ generates a $C_0$-semigroup in a Banach space $X$, $\xi \in X$, $g\in BV([ a,b ];X)$, and to study its main properties, such as regularity, compactness, and continuity. Some necessary and/or sufficient conditions for the compactness of the solution operator extending some earlier results due to the author and to Baras, Hassan, Veron, as well as some applications to the existence of certain generalized solutions to a semilinear equation involving distributed, or even spatial, measures, are also included. Two concrete examples of elliptic and parabolic partial differential equations subjected to impulsive dynamic conditions on the boundary illustrate the effectiveness of the abstract results.

References [Enhancements On Off] (What's this?)

  • N. U. Ahmed, Some remarks on the dynamics of impulsive systems in Banach Spaces, J. of Dynamics of Continuous, Discrete and Impulsive Systems Ser. A Math. Anal. 8 (2001), 261–274.
  • Pierre Baras, Jean-Claude Hassan, and Laurent Véron, Compacité de l’opérateur définissant la solution d’une équation d’évolution non homogène, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 14, A799–A802. MR 430864
  • V. Barbu, Personal communication.
  • V. Barbu and Th. Precupanu, Convexity and optimization in Banach spaces, 2nd ed., Mathematics and its Applications (East European Series), vol. 10, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1986. MR 860772
  • I. Bejenaru, J. I. Diaz, I. I. Vrabie, An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamic boundary conditions, Electronic Journal of Differential Equations, 2001, no. 50, 1-19.
  • S. Bochner, A. E. Taylor, Linear functionals on certain spaces of abstractly-valued functions, Ann. of Math. 39(1938), 913-944.
  • H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
  • Haïm Brézis and Avner Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. (9) 62 (1983), no. 1, 73–97. MR 700049
  • J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
  • Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
  • Bekkai Messirdi, Résonances des systèmes perturbeés d’opérateurs de Schrödinger, Maghreb Math. Rev. 5 (1996), no. 1-2, 113–121 (French, with English, French and Arabic summaries). MR 1801043
  • L. M. Graves, The Theory of Functions of Real Variables, McGraw-Hill Book Company, Inc. New York and London, 1946.
  • S. Gutman, Compact perturbations of $m$-accretive operators in general Banach spaces, SIAM J. Math. Anal. 13 (1982), no. 5, 789–800. MR 668321, DOI
  • E. Hille, R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloquium Publications, Volume 31, Fourth Printing of Revised Edition, 1981. (1st printing)
  • Norimichi Hirano, Local existence theorems for nonlinear differential equations, SIAM J. Math. Anal. 14 (1983), no. 1, 117–125. MR 686238, DOI
  • Norimichi Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc. 120 (1994), no. 1, 185–192. MR 1174494, DOI
  • Norimichi Hirano and Noriko Mizoguchi, Existence of periodic solutions for semilinear parabolic equations, Topology in nonlinear analysis (Warsaw, 1994) Banach Center Publ., vol. 35, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 39–49. MR 1448425
  • Chaim Samuel Hönig, Volterra Stieltjes-integral equations, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Functional analytic methods; linear constraints; Mathematics Studies, No. 16; Notas de Matemática, No. 56. [Notes on Mathematics, No. 56]. MR 0499969
  • A. Pazy, A class of semi-linear equations of evolution, Israel J. Math. 20 (1975), 23–36. MR 374996, DOI
  • A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
  • Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI
  • Naoki Shioji, Local existence theorems for nonlinear differential equations and compactness of integral solutions in $L^p(0,T;X)$, Nonlinear Anal. 26 (1996), no. 4, 799–811. MR 1362753, DOI
  • Naoki Shioji, Periodic solutions for nonlinear evolution equations in Banach spaces, Funkcial. Ekvac. 42 (1999), no. 2, 157–164. MR 1718787
  • Ioan I. Vrabie, The nonlinear version of Pazy’s local existence theorem, Israel J. Math. 32 (1979), no. 2-3, 221–235. MR 531265, DOI
  • Ioan I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), no. 3, 653–661. MR 1015686, DOI
  • Ioan I. Vrabie, A compactness criterion in $C(0,T;X)$ for subsets of solutions of nonlinear evolution equations governed by accretive operators, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), no. 1, 149–157. MR 859853
  • I. I. Vrabie, Compactness methods for nonlinear evolutions, 2nd ed., Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. With a foreword by A. Pazy. MR 1375237
  • Ioan I. Vrabie, Compactness in $L^p$ of the set of solutions to a nonlinear evolution equation, Qualitative problems for differential equations and control theory, World Sci. Publ., River Edge, NJ, 1995, pp. 91–101. MR 1372742

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47D06, 46G10, 47B07, 35A05, 35J99, 35K99

Retrieve articles in all journals with MSC (2000): 47D06, 46G10, 47B07, 35A05, 35J99, 35K99

Additional Information

Ioan I. Vrabie
Affiliation: Faculty of Mathematics, “Al. I. Cuza" University of Iaşi, Iaşi 6600, Romania
Address at time of publication: P. O. Box 180, Ro, Iş 1, Iaşi 6600, Romania

Keywords: Linear evolution equation, $C_0$-semigroup, vector-valued function of bounded variation, compactness of the solution operator
Received by editor(s): March 19, 2001
Received by editor(s) in revised form: September 21, 2001
Published electronically: April 1, 2002
Additional Notes: This research was supported in part by the CNCSU/CNFIS Grant C120(1998) of the World Bank and the Romanian Government
Article copyright: © Copyright 2002 American Mathematical Society