Higher integrability for parabolic equations of p(x, t)-laplacian type

Advances in Differential Equations

Volumn 10, Issue 9, 2005, Pages 1053-1080

  Antontsev, Stanislav ^a   Zhikov, Vasilii ^b  

^a UNIVERSITY OF BEIRA INTERIOR   (Portugal)

^b VLADIMIR STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 55349121945     PISSN: 10799389     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (34)

1. E. Acerbi and G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Rational Mech. Anal., 156 (2001), 121-140.
2. E. Acerbi, Regularity results for stationary electro-rheological fluids, Arch. Rational Mech. Anal., 164 (2002), 213-259.
3. E. Acerbi, G. Mingione, and G. Seregin, Regularity results for paraboliqc systems related to a class of non-Newtonian fluids, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 21(01) (2004), 25-60.
4. S. Antontsev, Nonlinear parabolic equations with a variable exponent of degeneration, Book of abstracts. International conference "Nonlinear partial differential equations", Ukraine, Alushta, September 15-21, 2003. Institute of Appliied Mathematics and Mechanics of NASU, Donetsk, (2003), p. 12.
5. S. N. Antontsev, J. I. Díaz, and H. B. Oliveira, Stopping a viscous fluid by a a feedback dissipative field: thermal effects without phase changing, Progress in Nonlinear Diferential Equations and Their Applications, (Proceedings TPDE, Obidos, Portugal, June 7-10, 2003), Birkhäuser, Verlag Basel/Switzerland, vol. 61, 1-14, 2005.
6. S. N. Antontsev, J. I. Díaz, and S. Shmarev, "Energy Methods for Free Boundary Problems:Applications to Non-linear PDEs and Fluid Mechanics," Bikhäuser, Boston, 2002. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48.
7. S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov, "Boundary value problems in mechanics of nonhomogeneous fluids," North-Holland Publishing Co., Amsterdam, 1990. Translated from original russian edition: Nauka, Novosibirsk, (1983).
8. S. N. Antontsev and F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara, Sez., Sc. Mat. To appear.
9. S. N. Antontsev and S. I. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Preprint, Universidade da Beira Interior, Portugal, 4 (2004), 1-38.
10. S. N. Antontsev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Journal of Nonlinear Analysis, 60 (2005), 515-545.
11. A. Arkhipova, On modifications of the Gehring lemma appearing in the study of parabolic initial-boundary value problems, Probl. Mat. Anal. 17, 259 (1997), 20-45.
12. A. Arkhipova and O. Ladyzhenskaya, On a generalization of Gehring's lemma, Boundary value problems of mathematical physics and related problems of function theory. Work collection. Dedicated to N. N. Ural'tseva. Sankt-Peterburg, Ladyzhenskaya, O. A. (ed.), 259 (1999), 7-18.
13. E. DiBenedetto, "Degenerate parabolic equations," Springer-Verlag, New York, 1993.
14. E. DiBenedetto, J. Urbano, and V. Vespri, Current issues on singular and degenerate evolution equations, Pré-Publica̧cões do Departamento de Matemática, Universidade de Coimbra, (2003), 1-127.
15. F. Duzaar and G. Mingione, Regularity for degenerate elliptic problems via pharmonic approximation, Ann. I.H. Poincaré-AN, 21 (2004), 735-766.
16. L. Esposito, F. Leonetti, and G. Mingione, Higher integrability for minimizers of integral functionals with (p, q) growth, J. Differential Equations, 157 (1999), 414-438.
17. X. Fan and D. Zhao, Regularity of minimizers of variational integrals with continuous p(x)-growth contditions, Chinese Ann. Math., 17A(5) (1996), 557-564.
18. X. Fan and D. Zhao, Regularity of minimizers of variational integrals with continuous p(x)-growth conditions, Chinese Journal of Contemporary Mathematics, 17 (4) (1996), 327-336.
19. X. Fan, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295-318.
20. X. Fan, The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear Anal., 39 (2000), 807-816.
21. 1,0 regularity of weak solutions for p(x)-Laplacian equations, J. Gansu Education College, 15(2) (2001), 1-5.
22. T. Iwaniec and C. Sbordone, Weak minima of variotional integrals, J. reine angew. Math., 454 (1994), 143-161.
23. J. Kinnunen and J. Lewis, Higher integrability for parabolic systems of p-Laplacian type, Duke Math. J, 102(2) (2000), 252-271.
24. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'tseva, "Linear and quasilinear equations of parabolic type," American Mathematical Society, Providence, R.I., 1967. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23.
25. p-estimate for the gradient of solutions of second order divergence equations, Ann. Sc. Norm. Sup. Pisa, 17 (1963), 189-206.
26. N. Meyers and A. Elcrat, Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J., 42 (1975), 121-136.
27. K. Rajagopal and M. Ružička, Mathematical modelling of electro-rheological fluids, Cont. Mech. Therm., 13 (2001), 59-78.
28. M. Ružička, "Electrorheological fluids: modeling and mathematical theory," Springer, Berlin, 2000. Lecture Notes in Mathematics, 1748.
29. N), Dokl. Ross. Acad. Nauk., 369(4) (1999), 451-454.
30. E. M. Stein, "Singular Integrals and Diferential Properties of Fuctions," Princeton University Press, Princeton, New Jersey, 1970.
31. E. Stredulinsky, Higher integrability from reverse Holder inequality, Indiana Univ. Math. J., 29 (1980), 407-413.
32. V. Zhikov, Meyers type estimates for the solution of a nonlinear Stokes system, Diferential Equations, 33 (1997), 108-115.
33. Nal of Mathematical Physics, 5 (1997), 105-116.
34. V. Zhikov, On density of smooth functions in Sobolev-Orlich spaces, Zapiski naychnux seminarov POMI, 310 (2004), 1-14.