Nonlinear parabolic equations with natural growth in general domains

A. Dall'aglio; D. Giachetti; J.-P. Puel

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 3, page 653-683
  • ISSN: 0392-4041

Abstract

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We prove an existence result for a class of parabolic problems whose principal part is the p -Laplace operator or a more general Leray-Lions type operator, and featuring an additional first order term which grows like | ∇ u | p . Here the spatial domain can have infinite measure, and the data may be not regular enough to ensure the boundedness of solutions. As a consequence, solutions are obtained in a class of functions with exponential integrability. An existence result of bounded solutions is also given under additional hypotheses.

How to cite

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Dall'aglio, A., Giachetti, D., and Puel, J.-P.. "Nonlinear parabolic equations with natural growth in general domains." Bollettino dell'Unione Matematica Italiana 8-B.3 (2005): 653-683. <http://eudml.org/doc/195326>.

@article{Dallaglio2005,
abstract = {We prove an existence result for a class of parabolic problems whose principal part is the $p$-Laplace operator or a more general Leray-Lions type operator, and featuring an additional first order term which grows like $|\nabla u |^\{p\}$. Here the spatial domain can have infinite measure, and the data may be not regular enough to ensure the boundedness of solutions. As a consequence, solutions are obtained in a class of functions with exponential integrability. An existence result of bounded solutions is also given under additional hypotheses.},
author = {Dall'aglio, A., Giachetti, D., Puel, J.-P.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {653-683},
publisher = {Unione Matematica Italiana},
title = {Nonlinear parabolic equations with natural growth in general domains},
url = {http://eudml.org/doc/195326},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Dall'aglio, A.
AU - Giachetti, D.
AU - Puel, J.-P.
TI - Nonlinear parabolic equations with natural growth in general domains
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/10//
PB - Unione Matematica Italiana
VL - 8-B
IS - 3
SP - 653
EP - 683
AB - We prove an existence result for a class of parabolic problems whose principal part is the $p$-Laplace operator or a more general Leray-Lions type operator, and featuring an additional first order term which grows like $|\nabla u |^{p}$. Here the spatial domain can have infinite measure, and the data may be not regular enough to ensure the boundedness of solutions. As a consequence, solutions are obtained in a class of functions with exponential integrability. An existence result of bounded solutions is also given under additional hypotheses.
LA - eng
UR - http://eudml.org/doc/195326
ER -

References

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  1. ARONSON, D. G. - SERRIN, J., Local behavior of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal.25 (1967), 81-122. Zbl0154.12001MR244638
  2. BLANCHARD, D. - PORRETTA, A., Nonlinear parabolic equations with natural growth terms and measure initial data, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 583-622. Zbl1072.35089MR1896079
  3. BOCCARDO, L. - MURAT, F. - PUEL, J.-P., Existence de solutions non bornées pour certaines équations quasi-linéaires, Portugal. Math., 41, n. 1-4 (1982), 507-534. Zbl0524.35041MR766873
  4. BOCCARDO, L. - MURAT, F. - PUEL, J.-P., Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl., (4) 152 (1988), 183-196. Zbl0687.35042MR980979
  5. BOCCARDO, L. - MURAT, F. - PUEL, J.-P., Existence results for some quasilinear parabolic equations, Nonlinear Anal., 13 (1989), 373-392. Zbl0705.35066MR987375
  6. BOCCARDO, L. - MURAT, F. - PUEL, J.-P., L 1 estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. (2), 23 (1992), 326-333. Zbl0785.35033MR1147866
  7. BOCCARDO, L. - MURAT, F. - PUEL, J.-P., Résultats d'existence pour certains problémes elliptiques quasilinéaires, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 213-235. Zbl0557.35051MR764943
  8. DALL'AGLIO, A. - DE CICCO, V. - GIACHETTI, D. - PUEL, J.-P., Existence of solutions for nonlinear elliptic equations in unbounded domains, Nonlinear Diff. Eq. and Appl., to appear. Zbl1120.35038MR2182422
  9. DALL'AGLIO, A. - GIACHETTI, D. - PUEL, J.-P., Nonlinear elliptic equations with natural growth in general domains, Ann. Mat. Pura Appl., to appear. Zbl1097.35050MR1939689
  10. DALL'AGLIO, A. - ORSINA, L., Nonlinear parabolic equations with natural growth conditions and L 1 data, Nonlin. Anal. TMA, 27, no. 1 (1996), 59-73. Zbl0861.35045MR1390712
  11. DI BENEDETTO, E., Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. Zbl0794.35090
  12. DONATO, P. - GIACHETTI, D., Quasilinear elliptic equations with quadratic growth on unbounded domains, Nonlin. Anal., 10 (1986), 791-804. Zbl0602.35036MR851147
  13. FERONE, V. - MURAT, F., Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonlinear Anal., 42 (2000), 1309-1326. Zbl1158.35358MR1780731
  14. FERONE, V. - POSTERARO, M. R. - RAKOTOSON, J.-M., L 1 -estimates for nonlinear elliptic problems with p-growth in the gradient, J. Ineq. Appl., 3 (1999), 109-125. Zbl0928.35060MR1733106
  15. FERONE, V. - POSTERARO, M. R. - RAKOTOSON, J.-M., Nonlinear parabolic equations with p-growth and unbounded data, C. R. Acad. Sci. Paris Sér. I Math., 328, no. 4 (1999), 291-296. Zbl0922.35075MR1675940
  16. FERONE, V. - POSTERARO, M. R. - RAKOTOSON, J.-M., Nonlinear Parabolic Problems with Critical Growth and Unbounded Data, Indiana Math. J.50, no. 3 (2001), 1201- 1215. Zblpre01780897MR1871353
  17. GRENON, N., Existence results for some quasilinear parabolic problems, Ann. Mat. Pura Appl., 4 (1993), 281-313. Zbl0806.35089MR1271423
  18. GRENON, N., Asymptotic behaviour for some quasilinear parabolic equations, Nonlinear Anal., 20 (1993), 755-766. Zbl0812.35065MR1214741
  19. LANDES, R. - MUSTONEN, V., On parabolic initial-boundary value problems with critical growth for the gradient, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 135-158. Zbl0836.35078MR1267364
  20. LERAY, J. - LIONS, J.L., Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97- 107. Zbl0132.10502MR194733
  21. LIONS, J.-L., Quelques mèthodes de rèsolution des problémes aux limites non linèaires, Dunod, Gauthier-Villars, Paris, 1969. Zbl0189.40603MR259693
  22. NICOLOSI, F., Weak solutions of boundary value problems for degenerate parabolic operators in unbounded open sets. Boll. Un. Mat. Ital.6-C, no. 1 (1985), 269-278. Zbl0594.35051MR805219
  23. ORSINA, L. - PORZIO, M.M., L ∞ ( Q ) -estimate and existence of solutions for some nonlinear parabolic equations, Boll. U.M.I.6-B (1992), 631-647. Zbl0783.35026MR1191957
  24. PUEL, J.-P., A compactness theorem in quasilinear parabolic problems and application to an existence result, Nonlinear parabolic equations: qualitative properties of solutions (Rome, 1985), Pitman Res. Notes Math. Ser., 149, Longman Sci. Tech., Harlow (1987), 189-199. Zbl0655.35035MR901109
  25. SIMON, J., Compact sets in the space L p ( 0 , T ; B ) , Ann. Mat. Pura Appl. (4) 146 (1987), 65-96. Zbl0629.46031MR916688
  26. STAMPACCHIA, G., Equations elliptiques du second ordre à coefficients discontinus. Séminaire de Mathématiques Supérieures, No. 16Les Presses de l'Université de Montréal, Montrèal, Que. (1966). Zbl0151.15501MR251373

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