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Representation theorems for parabolic systems

Part of: Representations of solutions Parabolic equations and systems

Published online by Cambridge University Press:  09 April 2009

J. Chabrowski
J. Chabrowski
Affiliation:
Department of Mathematics, The University of Queensland, St. Lucia, Queensland 4067, Australia
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Abstract

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The aim of this article is to review the progress made in the last few years in the representation theory of solutions of parabolic systems in the sense of Petrowskii.

MSC classification

Secondary: 35K30: Initial value problems for higher-order parabolic equations 35C15: Integral representations of solutions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 32 , Issue 2 , April 1982 , pp. 246 - 288
Copyright
Copyright © Australian Mathematical Society 1982

References

Aronson, D. G. (1968), ‘Non-negative solutions of linear parabolic equations,’ Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 22, 607–694.Google Scholar
Aronson, D. G., ‘Widder's Inversion Theorem and the initial distribution problem’, Siam J. Math. Anal., to appear.Google Scholar
Besala, P. (1972), ‘Function classes pertaining to differential inequalities of parabolic type in unbounded regions’, Ann. Polon. Math. 25, 281–291.CrossRefGoogle Scholar
Billingsley, B. (1979), Probability and measure (John Wiley, New York).Google Scholar
Chabrowski, J. (1967a), ‘Bemerkungen über Zeichen der Elemente der Matrix der Grundlösungen für parabolishe Systeme von partiellen Differentialgleichungen zweiter Ordnung’, Ann. Polon. Math. 19, 287–300.CrossRefGoogle Scholar
Chabrowski, J. (1967b), ‘Les solutions non-négatives d'un système parabolique d'équations’, Ann. Polon. Math. 19, 193–197.CrossRefGoogle Scholar
Chabrowski, J. (1970), ‘Les propriétés des solutions non négatives d'un systéme parabolique d'équations’, Ann. Polon. Math. 22, 223–231.CrossRefGoogle Scholar
Chabrowski, J. (1971), ‘Certaines propriétés des solutions non négatives d'un systéme parabolique d'équations’, Ann. Polon. Math. 24, 137–143.CrossRefGoogle Scholar
Chabrowski, J. (1974), ‘Representation theorems and Fatou theorems for parabolic systems in the sense of Pietrowskii’, Colloq. Math. 31, 301–314.CrossRefGoogle Scholar
Chabrowski, J. and Watson, N. A. (1980), ‘Properties of solution of weakly coupled parabolic systems’, to appear.Google Scholar
Dvoretzky, A. (1971), ‘A note on Hausdorff dimension functions’, Proc. London Math. Soc. (3) 33, 385–451.Google Scholar
Eidel'man, S. D. (1969), Parabolic systems (North Holland, Amsterdam).Google Scholar
Friedman, A. (1964), Partial differential equations of parabolic type (Prentice-Hall, Englewood Cliffs).Google Scholar
Hayman, W. K. and Kennedy, P. B. (1976), Subharmonic functions I (Academic Press, London).Google Scholar
Johnson, R. (1971), ‘Representation theorems and Fatou theorems for second-order linear parabolic differential equations’, proc. London Math. Soc. 23, 325–347.CrossRefGoogle Scholar
Kondrat'ev, V. A. and Eidel'man, S. D. (1974), ‘Positive solutions of linear partial differential equations’, Trudy Moskov. Mat. Obšč. 31, English transl.,Google Scholar
Trans. Moscow Math. Soc. 31, 81–148.Google Scholar
Krasnosel'skii, M. A., Rutickii, Ya. B. (1961), Convex functions and Orlicz spaces (P. Noordhoff Ltd., Groningen, The Netherlands).Google Scholar
Krźyzánski, M. (1964), ‘Sur les solutions non négatives de l'équation linéaire normale parabolique’, Rev. Roumaine Math. Pures Appl. (5) 9, 393–408.Google Scholar
Kufner, A., John, O., Fučik, S. (1977), Function spaces, (Noordhoff International Publishing, Leyden, Academia Publishing House of the Czechoslovak Academy of Science, Prague).Google Scholar
Milicer-Gruzewska, H. (1960), ‘Recherches sur les propriétés de la solution d'un systéme parabolique d'équations’, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (3) 4, 255–280.Google Scholar
Milicer-Gruzewska, H. (1963), ‘Quelques propriétés des solutions fondamentales du systéme parabolique d'équations’, Publ. Sém. Géom. Univ. Neuchatel, (1) 3, 3–9.Google Scholar
Rogers, C. A. (1970), Hausdorff measures (Cambridge University Press).Google Scholar
Rogers, C. A. and Taylor, S. J. (1961), ‘Functions continuous and singular with respect to a Hausdorff measure’, Mathematica 8, 1–31.Google Scholar
Stein, E. M. (1970), Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, New Jersey).Google Scholar
Szarski, J. (1975), ‘Strong maximum principle for non-linear parabolic differential functional inequalities in arbitrary domains’, Ann. Polon. Math. 31, 197–303.CrossRefGoogle Scholar
Watson, N. A. (1974), ‘Uniqueness and representation theorems for parabolic equations’, J. London Math. Soc. 8, 311–321.CrossRefGoogle Scholar
Watson, N. A. (1977), ‘Differentiation of measures and initial values of temperatures’, J. London Math. Soc. 16, 271–282.CrossRefGoogle Scholar
Watson, N. A. (1978), ‘The rate of special decay of non-negative solutions of linear parabolic equations’, Arch. Rational Mech. Anal. 68, 121–124.CrossRefGoogle Scholar
Watson, N. A. (1980), ‘Initial singularities of Gauss-Weierstrass integrals and their relations to Laplace transform and Hausdorff measures’, to appear.CrossRefGoogle Scholar
Widder, D. A. (1944), ‘Positive temperature on an infinite rod’, Trans. Amer. Math. Soc. 55, 85–95.CrossRefGoogle Scholar
Wilcox, C. H. (1980), ‘Positive temperatures with prescribed initial heat distributions’, Amer. Math. Monthly 87, 183–186.CrossRefGoogle Scholar