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Strong maximum principles for weakly coupled systems of quasilinear parabolic inequalities

Published online by Cambridge University Press:  09 April 2009

M. A. Dow
M. A. Dow
Affiliation:
Department of External Studies, University of Queensland, Brisbane, Australia
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Abstract

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Výborný and I (1972) proved maximum principles for a quasilinear elliptic operator where the boundary satisfied a smoothness condition weaker than the interior sphere property. In this paper I extend these to parabolic operators of a similar form and through a simple device to weakly coupled systems of such operators. Finally, I extend all of these results to an operator similar to the “parabolic” case of an operator introduced by Redheffer (1971). His conditions on the coefficients are replaced by conditions analogous to those Dow and Výborný (1972).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 1 , February 1975 , pp. 103 - 120
Copyright
Copyright © Australian Mathematical Society 1975

References

Ako, K. (1968), ‘Semi-decomposable systems and vector subfunctions’, Funkcialaj Ekvacioj 8, 91–97.Google Scholar
Bony, J.-M. (1969), ‘Principle du maximum, integalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés’, Ann. Inst. Fourier(Grenoble) 19, fasc. 1, 277–304.Google Scholar
Dow, M. A. (1972), Maximum principles for some quasilinear degenerate elliptic-parabolic operators of second order, Dissertation, (University of Queensland, 1972).Google Scholar
Dow, M. A. and Výborný, R. (1972), ‘Maximum principles for some quasilinear second order partial differential equations’, Rend. Sem. Mat. Univ. Padova 47, 331–351.Google Scholar
Hill, C. D. (1970), ‘A sharp maximum principle’, Indiana Univ. Math. J. 20, 213–229.CrossRefGoogle Scholar
Horáček, O. and Výborný, R. (1966), ‘Über eine fastlineare partielle Differentialgeichung vom nichtyperbolischen Typus’, Comment. Math. Univ. Carolinae 7, 3,261–264.Google Scholar
Il'in, A. M., Kalashnikov, A. C. and Oleinik, O. A. (1962), ‘Linear second order equations of parabolic type’, (Russian) Uspekhi Mat. Nauk 17, 1–146.Google Scholar
English transl. Russian Math. Surveys 17, 1–143.Google Scholar
Khusnytdinova, N. V. (1967), ‘The limiting moisture profile during infiltration into a homogeneous soil’, (Russian) Prikl. Mat. Mekh. 31, 770–776.Google Scholar
English transl. J. Appl. Math. Mech. 31, 783–789.Google Scholar
Kusano, T. (1963), ‘On the maximum principle for quasilinear parabolic equations of theh second order’, Proc. Japan. Acad. 39, 211–216.Google Scholar
Ladyzhenskaia, O. A., Solonikova, V. A. and Ural'tseva, N. N. (1968), Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs 23 (Amer. Math. Soc., 1968).CrossRefGoogle Scholar
Moisak, P. P. (1971), é's method for second order parabolic systems’, (Ukrainian) Dopovidi Akad. Nauk Ukrain. RSR Ser A, 109–113, 187.Google Scholar
Nirenberg, L. (1953), ‘A strong maximum principle for parabolic equations’, Comm. Pure Appl. Math. 6, 167–177.CrossRefGoogle Scholar
Protter, M. H. and Weinberger, H. F. (1967), Maximum principles in differential equations (Prentice Hall, 1967).Google Scholar
Pucci, C. (1957–1958), ‘Properitá di massimo e minimi delle soluzioni di equazioni a derivate parziali del secondo ordine di tipo ellittico e parabolico I, II’, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 23, 370–375; 24, 3–6.Google Scholar
Redheffer, R. M. (1962), ‘An extension of certain maximum principles’, Monatsh. Math. 66, 32–42.CrossRefGoogle Scholar
Redheffer, R. M. (1971), ’, Indiana Univ. Math. J. 21, 227–248.CrossRefGoogle Scholar
Stroock, D. W. (1970), ‘On certain system of parabolic equations’, Comm. Pure. Appl. Math., 23, 447–457.CrossRefGoogle Scholar
Výborný, R. (1958), ‘Some basic properties of solutions to boundary value problems for parabolic partial differential equations’, (Russian) Czechoslovak Math. J. 8 (83), 537–551.CrossRefGoogle Scholar
Výborný, R. (1963), ‘On certain extension of the maximum principle’, Differential Equations and Their Applications (Proc. Conf. Prague, 1962) pp. 223–228. (Publ. House Czechoslovak Acad. Sci., Prague; Academic Press, New York, 1963).Google Scholar
Wasowski, J. (1970), ‘Maximum principles for a certain strongly elliptics system of linear equations of second order’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18, 741–745.Google Scholar