Book Review
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MathSciNet review:
1568164
Full text of review:
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Book Information:
Author:
C. V. Pao
Title:
Nonlinear parabolic and elliptic equations
Additional book information:
Plenum Press, New York and London, 1992, xv+777 pp., US$125.00. ISBN 0-306-44343-0.
Oskar Perron, Ein neuer Existenzbeweis für die Integrale der Differentialgleichung $y\prime =f(x,y)$, Math. Ann. 76 (1915), no. 4, 471–484 (German). MR 1511836, DOI 10.1007/BF01458218
Mitio Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J. 6 (1954), 207–229. MR 70014
Kiyoshi Akô, On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan 13 (1961), 45–62. MR 147758, DOI 10.2969/jmsj/01310045
Herbert B. Keller, Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal. 35 (1969), 363–381. MR 255979, DOI 10.1007/BF00247683
Herbert Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 125–146. MR 296498, DOI 10.1512/iumj.1971.21.21012
Klaus Schmitt, Boundary value problems for quasilinear second-order elliptic equations, Nonlinear Anal. 2 (1978), no. 3, 263–309. MR 512661, DOI 10.1016/0362-546X(78)90019-6
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979–1000. MR 299921, DOI 10.1512/iumj.1972.21.21079
C. V. Pao, Successive approximations of some nonlinear initial-boundary value problems, SIAM J. Math. Anal. 5 (1974), 91–102. MR 339507, DOI 10.1137/0505010
C. V. Pao, Positive solutions of a nonlinear boundary-value problem of parabolic type, J. Differential Equations 22 (1976), no. 1, 145–163. MR 422876, DOI 10.1016/0022-0396(76)90008-5
J.-P. Puel, Existence, comportement à l’infini et stabilité dans certains problèmes quasilinéaires elliptiques et paraboliques d’ordre $2$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 1, 89–119. MR 399654
Howard A. Levine and Lawrence E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations 16 (1974), 319–334. MR 470481, DOI 10.1016/0022-0396(74)90018-7
Avner Friedman and Bryce McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), no. 2, 425–447. MR 783924, DOI 10.1512/iumj.1985.34.34025
Siegfried Carl, On the existence of extremal weak solutions for a class of quasilinear parabolic problems, Differential Integral Equations 6 (1993), no. 6, 1493–1505. MR 1235207
Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. MR 415432, DOI 10.1137/1018114
- [1]
- O. Perron, Ein Neuer Existenzbeweis fur sie Integrale der Differentialgleichung
, Math. Ann. 76 (1915), 471-484. MR 1511836
- [2]
- M. Nagumo, On principally linear elliptic differential equations of second order, Osaka. J. Math. 6 (1954), 207-229. MR 0070014 (16:1116a)
- [3]
- K. Ako, On the Dirichlet problem of quasi-linear elliptic differential equations of second order, J. Math. Soc. Japan 13 (1961), 45-62. MR 0147758 (26:5272)
- [4]
- H. B. Keller, Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal. 5 (1969), 363-381. MR 0255979 (41:639)
- [5]
- H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125-146. MR 0296498 (45:5558)
- [6]
- K. Schmitt, Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal. 2 (1978), 263-309. MR 512661 (80b:35064)
- [7]
- D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979-1000. MR 0299921 (45:8969)
- [8]
- C. V. Pao, Successive approximations of some nonlinear initial-boundary value problems, SIAM J. Math. Anal. 5 (1974), 91-102. MR 0339507 (49:4265)
- [9]
- -, Positive solutions of a nonlinear boundary value problem of parabolic type, J. Differential Equations 22 (1976), 145-163. MR 0422876 (54:10862)
- [10]
- J. P. Puel, Existence, comportement à l'infini et stabilité dans certaines problèmes quasilinéaires elliptiques et paraboliques d'ordre 2, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 3 (1976), 89-119. MR 0399654 (53:3497)
- [11]
- H. A. Levine and L. E. Payne, Nonexistence theorem for the heat equation with nonlinear boundary conditions and for porous medium equation backward in time, J. Differential Integral Equations 16 (1974), 319-334. MR 0470481 (57:10235)
- [12]
- A. Friedman and B. McLeod, Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447. MR 783924 (86j:35089)
- [13]
- S. Carl, On existence of extremal weak solutions for a class of quasilinear parabolic problems, Differential Integral Equations 6 (1993), 1493-1505. MR 1235207 (94h:35124)
- [14]
- H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. MR 0415432 (54:3519)
Review Information:
Reviewer:
Jerrold Bebernes
Journal:
Bull. Amer. Math. Soc.
32 (1995), 166-169
DOI:
https://doi.org/10.1090/S0273-0979-1995-00568-9
