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Spatial critical points of solutions of a one-dimensional nonlinear parabolic problem


Author: Lawrence Turyn
Journal: Proc. Amer. Math. Soc. 106 (1989), 1003-1009
MSC: Primary 35K55; Secondary 35B05, 58E05
DOI: https://doi.org/10.1090/S0002-9939-1989-0961417-5
MathSciNet review: 961417
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Abstract: The number of spatial critical points is nonincreasing in time, for positive, analytic solutions of a scalar, nonlinear, parabolic partial differential equation in one space dimension. While proving this, we answer the question: What happens to a critical point which loses simplicity?


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  • [1] S. B. Angenent, The Morse-Smale property for a semi-linear parabolic equation, J. Differential Equations 62 (1986), 427-442. MR 837763 (87e:58115)
  • [2] P. Brunovsky and B. Fiedler, Simplicity of zeros in scalar parabolic equations, J. Differential Equations 62 (1986), 237-241. MR 833419 (88c:35079)
  • [3] J. R. Cannon, The one-dimensional heat equation, Addison-Wesley, Menlo Park, California, 1984. MR 747979 (86b:35073)
  • [4] S.-N. Chow and J. K. Hale, Methods of bifurcation theory, Springer-Verlag, New York, 1982. MR 660633 (84e:58019)
  • [5] A. Friedman, On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations, J. Math. Mech. 7 (1958), 43-59. MR 0118970 (22:9739)
  • [6] D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations 59 (1985), 165-205. MR 804887 (86m:58080)
  • [7] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 401-441. MR 672070 (84m:35060)
  • [8] W-M. Ni and P. Sacks, The number of peaks of positive solutions of semilinear parabolic equations, SIAM J. Math. Anal. 16 (1985), 460-471. MR 783973 (87b:35084)
  • [9] K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, J. Reine Angew. Math. 211 (1962), 78-94. MR 0146534 (26:4056)
  • [10] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, N.J., 1967. MR 0219861 (36:2935)
  • [11] P. C. Rosenbloom and D. V. Widder, Expansions in terms of heat polynomials and associated functions, Trans. Amer. Math. Soc. 92 (1959), 220-266. MR 0107118 (21:5845)
  • [12] W. Walter, Differential and integral inequalities, Springer-Verlag, New York, (1970). MR 0271508 (42:6391)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0961417-5
Article copyright: © Copyright 1989 American Mathematical Society

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