Nonlinear parabolic differential equations and inequalities

Discrete and Continuous Dynamical Systems

Volumn 8, Issue 2, 2002, Pages 451-468

  Walter, Wolfgang ^a  

^a UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

Minimum principles; Nonlinear inequalities; p Laplacian; Parabolic differential equations; Radial solutions

Indexed keywords

EID: 23044533952     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2002.8.451     Document Type: Article

References (17)

1. A. Friedman, Remarks on the maximum principle for parabolic equations and its application. Pacific J. Math. 8 (1958), 201-211.
2. A. Friedman, B. McLeod, Blow-up of solutions of nonlinear degenerate parabolic equations. Arch. Rat. Mech. Anal. 96 (1986), 55-80.
3. C. F. Gaußs, Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des quadrats der Entfernung wirkenden Anziehungs - und Abstoßsungskräfte, Bcob. d. magn. Vereins f. 1839. Ges.Werke 5, pp. 200-242.
4. Lehrsatz 20 (p. 222) is the mean value theorem for harmonic functions
5. Lehrsatz 21 (p. 223) can be interpreted as a version of the strong maximum-minimum principle.
6. E. Hopf, Elementare Bemerkungen über die Lösungen partieller differentialgleichungen zweiter ordnung vom elliptischen typus, Sitz.-ber. Preuss. Akad. Wiss. 19 (1927), 147-152.
7. E. Hopf, A remark on linear elliptic differential equations of second order, Proc. Amer. Math. Soc., vol. 3, 1952, pp. 791-793.
8. M. Nagumo, Note in "Kansū-Hōteisiki" No. 15, 1939. (Japanese. Reference given in Nagumo-Simoda, 1951.)
9. L. Nirenberg, A strong maximum principle for parabolic equations, Commun. Pure Appl. Math. 6 (1953), 167-177.
10. O. A. Oleinik, On properties of solutions of certain boundary problems for equations of elliptic type, Mat. Sbornik, N.S. 30 (1952), 695-702.
11. M. A. Paraf, Sur le probléme de Dirichlet et son extension au cas de l'équation linéaire générale du second ordre, Ann. Fac. Sci. Toulouse 6, Fasc. 47 - Fasc. 54, 1892.
12. R. Vyborny, The properties of the solutions of certain boundary problems for parabolic equations, Dokl. Akad. Nauk SSSR 117 (1957), 563-565.
13. W. Walter, Differential- und integral-ungleichungen, Springer Tracts in Natural Philosophy, vol. 2, Springer-Verlag, Heidelberg, 1964.
14. W. Walter, Differential and integral inequalities, Ergebnisse der Mathematik, Bd. 55, Springer-Verlag, Heidelberg, 1970.
15. W. Walter, On the strong maximum principle for parabolic differential equations, Proc. Edinburgh Math. Society, vol. 29, 1986, pp. 93-96.
16. W. Walter, Differential inequalities. Inequalities - fifty years on from Hardy, Littelwood, and Polya, Marcel Dekker, New York, 1991, pp. 249-281.
17. W. Walter, Differential inequalities and maximum principles; theory, new methodes and applications, Nonlinear Analysis 30 (1997), 4695-4711.