Hostname: page-component-6bb9c88b65-zjgpb Total loading time: 0 Render date: 2025-07-27T21:22:42.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear analytic semiflows*

Published online by Cambridge University Press:  14 November 2011

Sigurd B. Angenent
Sigurd B. Angenent
Affiliation:
Department of Mathematics, University of Wisconsin at Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems (x′(t) = f(x(t))), and quasilinear initial value problems (x′(t)=A(x(t))x(t) + f(x(t))). This theory extends the theory of DaPrato and Grisvard of 1979, and shows how various properties, like analyticity of solutions, can be derived as a direct corollary of the existence theorem.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 1-2 , 1990 , pp. 91 - 107
Copyright
Copyright © Royal Society of Edinburgh 1990

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.)

Article purchase

Temporarily unavailable

References

1Amann, H.. Quasilinear evolution equations and parabolic systems. Trans. Amer. Math. Soc. 293 (1986), 191–227.CrossRefGoogle Scholar
2Amann, H.. On abstract parabolic fundamental solutions. J. Math. Soc. Japan 39 (1987), 96–116.CrossRefGoogle Scholar
3Amann, H.. Quasilinear parabolic systems under nonlinear boundary conditions. Arch. Rational Mech. Anal. 92 (1986), 153–192.CrossRefGoogle Scholar
4Angenent, S. B.. Abstract parabolic initial value problems (Leiden: University of Leiden, Report 22, 1986).Google Scholar
5Angenent, S. B.. Analyticity of the interface of the porous media equation after the waiting time. Proc. Amer. Math. Soc. 102 (1988), 329–336.Google Scholar
6Angenent, S. B.. Local existence and regularity for a class of degenerate parabolic equations. Math. Ann. 280 (1988), 465–482.CrossRefGoogle Scholar
7Angenent, S. B.. Large time asymptotics for the porous media equation. In Nonlinear Diffusion Equations and their Equilibrium States I, eds. Ni, W. M., Peletier, L. A. and Serrin, J., MSRI publication 12 (Berlin: Springer, 1988).CrossRefGoogle Scholar
8Angenent, S. B. and Gurtin, M. E., Multiphase thermomechanics with interfacial structure - 2. Evolution of an isothermal interface. Arch. Rational Mech. Anal. (to appear).Google Scholar
9Angenent, S. B. and Aronson, D. G. (in preparation).Google Scholar
10Arnold, V. I.. Geometrical Methods in the Theory of Ordinary Differential Equations, Grundlehren 250 (New York: Springer, 1983).Google Scholar
11Baillon, J. B.. Caractère borné de certains générateurs de semigroups linéaires dans les espaces de Banach. C. R. Acad. Sci. Paris 290 (1980), 757–760.Google Scholar
12Bergh, J. and Löfström, J.. Interpolation spaces, an introduction (Berlin: Springer 1976).Google Scholar
13Butzer, P. L. and Berens, H.. Semigroups of operators and approximation (Berlin: Springer, 1967).CrossRefGoogle Scholar
14Clément, Ph., Heijmans, H. J. A. M. et al. One Parameter Semigroups, CWI Monographs (Amsterdam: North Holland, 1987).Google Scholar
15DaPrato, G. and Grisvard, P.. Sommes d'opérateurs linéaires et équations différentielles opérationelles. J. Math. Pures Appl. 54 (1975), 305–387.Google Scholar
16DaPrato, G. and Grisvard, P.. Équations d'évolutions abstraites nonlinéair de type parabolique. Ann. Mat. Pura Appl. 120 (1979), 329–396.Google Scholar
17Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (New York: Springer 1981).CrossRefGoogle Scholar
18Lunardi, A.. Analyticity of the maximal solution of an abstract parabolic equation. Nonlinear Anal. 6 (1982), 503–521.CrossRefGoogle Scholar
19Lunardi, A.. Global solutions of abstract quasilinear problems. J. Differential Equations 58 (1985), 228–242.CrossRefGoogle Scholar
20Lunardi, A.. Quasilinear parabolic equations. Math. Ann. 267 (1984), 395–415.CrossRefGoogle Scholar
21Pazy, A.. Semigroups of linear operators and applications to partial differential equations. (New York: Springer, 1983).CrossRefGoogle Scholar
22Shub, M.. Global Stability of Dynamical Systems (New York: Springer, 1987).Google Scholar
23Triebel, H.. Interpolation theory, function spaces, differential operators (Amsterdam: North Holland, 1978).Google Scholar
24Sinestrari, E.. On the abstract Cauchy problem of parabolic type in spaces of continuous functions. J. Math. Anal. Appl. 107 (1985), 16–66.Google Scholar