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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Degenerate parabolic equations
with initial data measures

Author: Daniele Andreucci
Journal: Trans. Amer. Math. Soc. 349 (1997), 3911-3923
MSC (1991): Primary 35K65, 35R05; Secondary 35K55, 35K15
MathSciNet review: 1333384
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Abstract: We address the problem of existence of solutions to degenerate (and nondegenerate) parabolic equations under optimal assumptions on the initial data, which are assumed to be measures. The requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation, and with the form of the nonlinear forcing term. The latter depends on the space gradient of a power of the solution. Applications to related problems are also outlined.

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  • 1. H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis, Texte zur Math., 133, eds. H. Schmeisser and H. Triebel, Teubner, Stuttgart-Leipzig, 1993, 9-126. MR 94m:35153
  • 2. D. Andreucci, New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources, Manuscripta Math. 77 (1992), 127-159. MR 93k:35129
  • 3. D. Andreucci, E. DiBenedetto, A new approach to initial traces in nonlinear filtration, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 305-334. MR 92g:35089
  • 4. D. Andreucci, E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Scuola Norm. Sup. Pisa 18 (1991), 363-441. MR 92m:35146
  • 5. D. G. Aronson, L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. AMS 280 (1983), 351-366. MR 85c:25042
  • 6. P. Baras, M. Pierre, Critère d'existence de solutions positives pour des équations semilinéaires non monotones, Ann. Inst. H. Poincaré Anal. Non linéaire, 2 (1985), 185-212. MR 87j:45032
  • 7. Ph. Bènilan, M. G. Crandall, M. Pierre, Solutions of the porous medium equation in $\mathbf {R}^N$ under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), 51-87. MR 86b:35084
  • 8. L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169. MR 92d:35286
  • 9. E. DiBenedetto, A. Friedman, Hölder estimates for non-linear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22. MR 87f:35134a
  • 10. Y. Giga, T. Miyakawa, Navier-Stokes flow in $\mathbf {R}^3$ with measures as initial vorticity and Morrey spaces, Comm. PDE 14 (1989), 577-618. MR 90e:35130
  • 11. A. L. Gladkov, The Cauchy problem in classes of increasing functions for the equation of filtration with convection, Mat. Sbornik 186 (1995), 35-56 (transl. Sbornik Math. 186 (1995), 803-825). MR 96f:35093
  • 12. H. Hoshino, Y. Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac., 34 (1991), 475-494. MR 93c:34131
  • 13. K. Kobayasi, Semilinear parabolic equations with nonmonotone nonlinearity, Memoirs Sagami Inst. of Technology, 23 (1989), 83-99.
  • 14. H. Kozono, M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. PDE 19 (1994), 959-1014. MR 95d:35077
  • 15. O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural'tzeva, Linear and quasilinear equations of parabolic type, Transl. Math. Monos., 23, AMS, Providence, R.I., 1967. MR 39:3159b
  • 16. Y. Niwa, Semilinear heat equations with measures as initial data, Thesis, University of Tokyo, 1986.
  • 17. M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. PDE 17 (1992), 1407-1456. MR 94b:35218

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Additional Information

Daniele Andreucci
Affiliation: Università “La Sapienza”, Dipartimento di Metodi e Modelli Matematici, via A. Scarpa 16, 00161 Roma, Italy

Received by editor(s): August 15, 1994
Received by editor(s) in revised form: March 20, 1995
Additional Notes: The author is a member of GNFM of Italian CNR. Work supported by MURST project “Problemi non lineari...”
Article copyright: © Copyright 1997 American Mathematical Society

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