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Transactions of the American Mathematical Society

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A singular semilinear equation in $ L\sp{1}({\bf R})$


Authors: Michael G. Crandall and Lawrence C. Evans
Journal: Trans. Amer. Math. Soc. 225 (1977), 145-153
MSC: Primary 34B15
DOI: https://doi.org/10.1090/S0002-9947-1977-0477240-0
MathSciNet review: 0477240
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Abstract: Let $ \beta $ be a positive and nondecreasing function on R. The boundary-value problem $ \beta (u) - u'' = f,u'( \pm \infty ) = 0$ is considered for $ f \in {L^1}({\mathbf{R}})$. It is shown that this problem can have a solution only if $ \beta $ is integrable near $ - \infty $, and that if this is the case, then the problem has a solution exactly when $ \smallint _{ - \infty }^\infty f(x)dx > 0$.


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

  • [1] Ph. Benilan, H. Brezis and M. G. Crandall, A semilinear elliptic equation in $ {L^1}({{\mathbf{R}}^N})$, MRC Technical Summary Report #1526; Ann. Scuola Norm. Sup. Pisa (4) 2 (1975), 521-555. MR 0390473 (52:11299)
  • [2] H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espace de Hilbert, North-Holland Studies, no. 5, North-Holland, Amsterdam, 1973. MR 50 #1060. MR 0348562 (50:1060)
  • [3] H. Brezis and W. Strauss, Semi-linear second-order elliptic equations in $ {L^1}$, J. Math. Soc. Japan 25 (1973), 565-590. MR 49 #826. MR 0336050 (49:826)
  • [4] M. G. Crandall, An introduction to evolution governed by accretive operators, Dynamical Systems: An International Symposium, Vol. 1, L. Cesari, J. K. Hale, J. P. LaSalle, Editors, Academic Press, New York, 1976, 131-165. MR 0636953 (58:30550)
  • [5] T. Kurtz, Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc. 186 (1973), 259-272. MR 49 # 1256. MR 0336482 (49:1256)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0477240-0
Keywords: Nonlinear boundary value problem, accretive operator
Article copyright: © Copyright 1977 American Mathematical Society

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