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A semigroup treatment of a one-dimensional nonlinear parabolic equation


Author: Hisamitsu Serizawa
Journal: Proc. Amer. Math. Soc. 106 (1989), 187-192
MSC: Primary 35K55; Secondary 35K15, 47H06, 47H15, 58D07, 58D25
DOI: https://doi.org/10.1090/S0002-9939-1989-0953746-6
MathSciNet review: 953746
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the existence, uniqueness and differentiability of solutions (in a certain generalized sense) for the nonlinear parabolic equation

$\displaystyle {u_t} = {u_{xx}} - F(u,{u_x})(0 < x < 1,t > 0),$

under the maximal monotone boundary conditions:

$\displaystyle {( - 1)^i}{u_x}(t,i) \in {\beta _i}(u(t,i)),t > 0,i = 0,1.$


References [Enhancements On Off] (What's this?)

  • [1] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publ., Leyden, The Netherlands, Bucaresti, 1976. MR 0390843 (52:11666)
  • [2] M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265-298. MR 0287357 (44:4563)
  • [3] J. A. Goldstein and C. Y. Lin, Singular nonlinear parabolic boundary value problems in one space dimension, J. Differential Equations 68 (1987), 429-443. MR 891337 (89e:35079)
  • [4] P. Hartman, Ordinary differential equations, Wiley-Interscience, New York, 1964. MR 0171038 (30:1270)
  • [5] Y. Konishi, On $ {u_t} = {u_{xx}} - F({u_x})$ and the differentiability of the nonlinear semigroup associated with it, Proc. Japan Acad. 48 (1972), 281-286. MR 0322333 (48:695)
  • [6] S. Ôharu, A note on the generations of nonlinear semigroups in a locally convex space, Proc. Japan Acad. 43 (1967), 847-851. MR 0229092 (37:4670)
  • [7] H. Serizawa, $ M$-Browder-accretiveness of a quasilinear differential operator, Houston J. Math. 10 (1984), 147-152. MR 736582 (85h:35126)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0953746-6
Keywords: Nonlinear boundary value problem, $ m$-dissipative operator
Article copyright: © Copyright 1989 American Mathematical Society

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