Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-14T05:58:17.123Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear evolution equations in two Banach spaces

Published online by Cambridge University Press:  14 November 2011

Niko Sauer
Niko Sauer
Affiliation:
Department of Applied Mathematics, University of Pretoria, Pretoria, South Africa
Get access Rights & Permissions [Opens in a new window]

Synopsis

We consider the ordinary differential equation [Bu(t)]′ = Au(t) with A and B linear operators with domains in a Banach space X and ranges in a Banach space Y. The initial condition is that the limit as (t → 0 of Bu(t) is prescribed in Y. We study the properties of the “solution operator” S(t) which maps the initial state in Y to the solution u(t) at time t. The notion of infinitesimal generator A of S(t) is introduced and the relationships between S(t), an associated semi-group E(t), the operators A and B and some other operators are studied. In particular a pair of operators Ao and Bo, derived from A and B, determine the family S(t) of operators. These so-called “generating pairs” are characterized. The operators A and B and Ao and Bo need not be closed, but form so-called closed pairs which is a weaker condition. We also discuss two applications of the theory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 91 , Issue 3-4 , 1982 , pp. 287 - 303
Copyright
Copyright © Royal Society of Edinburgh 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Brezis, H.. On some degenerate nonlinear parabolic equations. Sympos. Pure Math. 18 (1970), 23–38.Google Scholar
2Brill, H.. A semilinear Sobolev equation in a Banach space. J. Differential Equations 24 (1977), 412–245.CrossRefGoogle Scholar
3Carroll, R. W. and Showalter, R. E.. Singular and degenerate Cauchy problems (New York: Academic Press, 1976).Google Scholar
4Dunford, N. and Schwartz, J. T.. Linear operators. Part I (New York: Interscience, 1964).Google Scholar
5Favini, A.. Laplace transform method for a class of degenerate evolution problems. Rend. Mat. 12 (1979), 511–536.Google Scholar
6Friedman, A.. Partial differential equations (New York: Holt, 1969).Google Scholar
7Fuchs, L.. Infinite abelian groups, Vol. I (New York: Academic Press, 1970).Google Scholar
8Fulton, C. T.. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 293–308.CrossRefGoogle Scholar
9Galpern, S. S.. Cauchy's problem for an equation of S. L. Sobolev's type. Dokl. Akad. Nauk SSSR 104 (1960), 640–643.Google Scholar
10Grabmüller, H.. Diskrete Approximation von Evolutionsgleichungen in Hilbertschen Rdumen (Doctoral dissertation, Univ. of Darmstadt, 1972).Google Scholar
11Krein, S. G.. Linear differential equations in Banach space. Translations of Mathematical Monographs, Vol. 29 (Providence, R.I.: Amer. Math. Soc, 1971).Google Scholar
12Lions, J. L., Boundary value problems. Technical Report, Univ. of Kansas, 1957.Google Scholar
13Lions, J. L.. Equations differentielles operationelles et problemes aux limites (Berlin: Springer Verlag, 1961).CrossRefGoogle Scholar
14Rossouw, W. J.. Conservation law formulations for boundary value problems. Pretoria Univ. Dept. Appl. Math. Res. Report UPTW6, 1977.Google Scholar
15Sandgren, L.. A vibration problem. Medd. Lunds Univ. Mat. Sem. 13 (1955), 1–84.Google Scholar
16Sauer, N.. Linear evolution equations in two Banach spaces. Pretoria Univ. Dept. Appl. Math. Res. Report UPTW17, 1980.Google Scholar
17Showalter, R. E.. Partial differential equations of Sobolev–Galpern type. Pacific J. Math. 31 (1969), 787–794.CrossRefGoogle Scholar
18Showalter, R. E. and Ting, T. W.. Pseudo-parabolic partial differential equations. SIAM J. Math. Anal. 1 (1970), 214–231.CrossRefGoogle Scholar
19Sobolev, S. L.. Some new problems in mathematical physics. Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), 3–50.Google Scholar
20Ting, T. W.. Parabolic and pseudo-parabolic partial differential equations. J. Math. Soc. Japan 21 (1969), 440–453.Google Scholar
21Yosida, K.. Functional analysis (Berlin: Springer Verlag, 1965).Google Scholar