Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-18T07:50:41.353Z Has data issue: false hasContentIssue false
  • English
  • Français

Semilinear evolution equations and fractional powers of a closed pair of operators

Published online by Cambridge University Press:  14 November 2011

Marié Grobbelaar-Van Dalsen
Marié Grobbelaar-Van Dalsen
Affiliation:
University of South Africa, P.O. Box 392, 0001 Pretoria, South Africa
Get access Rights & Permissions [Opens in a new window]

Synopsis

The nonlinear evolution problem [Bu(t)]′ = A(t, Bu)u + f(t, Bu) with B a constant linear operator and A = A(t, Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers of A and B. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 105 , Issue 1 , 1987 , pp. 101 - 115
Copyright
Copyright © Royal Society of Edinburgh 1987

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

1Agmon, S., Doughs, A. and Nirenberg, L.. Estimates for the Boundary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions I. Comm. Pure Appl. Math. 12 (1959), 623–727.CrossRefGoogle Scholar
2Brezis, H.. Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (Amsterdam: North Holland, 1973).Google Scholar
3Brill, H.. A semilinear evolution equation in a Banach space. J. Differential Equations 24 (1977), 412–425.CrossRefGoogle Scholar
4Brill, H.. Cauchy-Probleme fiir nichtlineare parabolische und pseudoparabolische Gleichungen. Dissertation, Ruhr-Universität Bochum, Germany, December 1974.Google Scholar
5Friedman, A.. Partial Differential Equations (New York: Holt, 1969).Google Scholar
6Friedman, A.. Remarks on nonlinear parabolic equations. Proc. Sympos. Appl. Math, 17 (1965), 3–23.CrossRefGoogle Scholar
7Fulton, C. T.. Two-point boundary value problems with eigenvalue parameter contained in the boundaryconditions. Proc. Roy. Soc. Edinburgh Sect A. 77 (1977), 293–308.CrossRefGoogle Scholar
8Gajewski, H. and Zacharias, K.. Über eine klasse nichtlinearer differentialgleichungen im Hilbertraum. J. Math. Anal. Appl. 44 (1973), 71–87.CrossRefGoogle Scholar
9Grange, O. and Mignot, F.. Sur la resolution d'une equation et d'une inequation paraboliques non linéaires. J. Fund. Anal. 11 (1972), 77–92.CrossRefGoogle Scholar
10Dalsen, M. Grobbelaar-Van. An Evolution Problem Involving non-Stationary Operators between two Banach Spaces. Quaestiones Math. 8 (1985), 97–130; 199–230.CrossRefGoogle Scholar
11Dalsen, M. Grobbelaar-Van. Fractional powers of a closed pair of operators. Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), 149–158.CrossRefGoogle Scholar
12Kato, T.. Fractional powers of Dissipative Operators, J. Math. Soc. Japan 13 (1961), 246–274.CrossRefGoogle Scholar
13Lions, J. L. and Magenes, E.. Non-Homogeneous Boundary Value Problems and Applications I (Berlin: Springer, 1972).Google Scholar
14Sauer, N.. Linear Evolution Equations in two Banach Spaces. Proc. Roy. Soc. Edinburgh Sect. A 91 (1982), 287–303.CrossRefGoogle Scholar
15Schauder, J.. Der Fixpunktsatz in Funktionalraumen. Studia Math. 2 (1930), 171–180.CrossRefGoogle Scholar
16Showalter, R. E.. Degenerate Evolution Equations and Applications. Indiana Univ. Math. J. 23 (1974), 655–677.CrossRefGoogle Scholar
17Showalter, R. E. and Carroll, R. W.. Singular and Degenerate Cauchy Problems (New York: Academic Press, 1976.)Google Scholar
18Showalter, R. E.. Nonlinear Degenerate Evolutions Equations and Partial Differential Equations of Mixed Type. SIAM J. Math. Anal. 6 (1975), 25–42.Google Scholar
19Sobolevski, P. E.. On Equations of Parabolic Type in a Banach space. Trudy Moskov. Mat. Obshch. 10 (1961), 297–350.Google Scholar
20Dalsen, M. van. Die Teorie van Nie-stasionere Evolusies geassosieer met Dinamies-Gekoppelde Randwaardeprobleme. Doctoral Thesis, Pretoria University, 1978.Google Scholar
21Yosida, K.. Fractional Powers of Infinitesimal Generators and Analyticity of the Semigroup generated by them. Proc. Japan Acad. Ser. A Math. Sci. 36 (1960), 86–89.CrossRefGoogle Scholar
22Yosida, K.. Functional Analysis (Berlin: Springer, 1980).Google Scholar