Regularity of solutions of parabolic equations with a double nonlinearity and a weight

M. D. Surnachëv

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 2, pp. 309-334.

Résumé

We study local regularity of solutions of nonlinear parabolic equations with a double degeneracy and a weight. We impose the condition of $ p$-admissibility on the weight; in particular this allows weights in the Muckenhoupt classes $ A_p$. We prove that solutions are locally Hölderian without any restriction on the sign being constant. We prove a Harnack inequality for nonnegative solutions. We examine the stability of the constants as the parameters in the equation approach the linear case.

Bibliographie

[1] Ivanov A. V., “Gelderovskie otsenki dlya kvazilineinykh parabolicheskikh uravnenii s dvoinym vyrozhdeniem”, Zap. nauchn. sem. LOMI, 171, 1989, 70–105 | Zbl

[2] Ivanov A. V., “Ravnomernye gelderovskie otsenki dlya obobschennykh reshenii kvazilineinykh parabolicheskikh uravnenii, dopuskayuschikh dvoinoe vyrozhdenie”, Algebra i analiz, 3:2 (1991), 139–179 | Zbl

[3] Ivanov A. V., “Klassy $\mathcal{B}_{m,l}$ i gelderovskie otsenki dlya kvazilineinykh parabolicheskikh uravnenii, dopuskayuschikh dvoinoe vyrozhdenie”, Zap. nauchn. sem. LOMI, 197, 1992, 42–70

[4] Ivanov A. V., “Kvazilineinye parabolicheskie uravneniya, dopuskayuschie dvoinoe vyrozhdenie”, Algebra i analiz, 4:6 (1992), 114–130 | Zbl

[5] Ivanov A. V., “Gelderovskie otsenki dlya uravnenii tipa bystroi diffuzii”, Algebra i analiz, 6:4 (1994), 101–142

[6] Ivanov A. V., “Gelderovskie otsenki dlya estestvennogo klassa uravnenii tipa bystroi diffuzii”, Zap. nauchn. sem. POMI, 229, 1995, 29–62 | Zbl

[7] Ivanov A. V., “Otsenki maksimumov modulei obobschennykh reshenii dlya dvazhdy nelineinykh parabolicheskikh uravnenii”, Zap. nauchn. sem. POMI, 221, 1995, 83–113

[8] Ivanov A. V., “The regularity theory for $(m,l)$-Laplacian parabolic equation”, Zap. nauchn. sem. POMI, 243, 1997, 87–110 | MR | Zbl

[9] Zhikov V. V., “O vesovykh sobolevskikh prostranstvakh”, Matem. sb., 189:8 (1998), 27–58 | DOI | MR | Zbl

[10] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR

[11] Bonafede S., Skrypnik I. I., “On Hölder continuity of solutions of doubly nonlinear parabolic equations with weight”, Ukr. matem. zh., 51:7 (1999), 890–903 | MR | Zbl

[12] Chiarenza F., Frasca M., “Boundedness for the solutions of a degenerate parabolic equation”, Appl. Anal., 17 (1984), 243–261 | DOI | MR | Zbl

[13] Chiarenza F. M., Serapioni R. P., “A Harnack inequality for degenerate parabolic equations”, Comm. Part. Diff. Eq., 9:8 (1984), 719–749 | DOI | MR | Zbl

[14] Chiarenza F., Serapioni R., “Degenerate parabolic equations and Harnack inequality”, Ann. Mat. Pura Appl. (4), 137 (1984), 139–162 | DOI | MR | Zbl

[15] Chiarenza F., Serapioni R., “A remark on a Harnack inequality for degenerate parabolic equations”, Rend. Sem. Mat. Univ. Padova, 73 (1985), 179–190 | MR | Zbl

[16] Chiarenza F., Serapioni R., “Pointwise estimates for degenerate parabolic equaitions”, Appl. Anal., 23:4 (1987), 287–299 | DOI | MR | Zbl

[17] Colding T. H., Minicozzi W. P. (II), “Liouville theorems for harmonic sections and applications”, Comm. Pure. Appl. Math., 51:2 (1998), 113–138 | 3.0.CO;2-E class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[18] DiBenedetto E., Degenerate parabolic equations, Springer-Verlag, New York, 1993 | MR

[19] DiBenedetto E., Gianazza U., Vespri V., “Harnack estimates for quasi-linear degenerate parabolic differential equations”, Acta Math., 200:2 (2008), 181–209 | DOI | MR | Zbl

[20] DiBenedetto E., Gianazza U., Vespri V., Harnack's inequality for degenerate and singular parabolic equations, Springer, New York, 2012 | MR

[21] Heinonen J., Kilpeläinen T., Martio O., Nonlinear potential theory of degenerate elliptic equations, Dover Publ. Inc., Mineola, NY, 2006 | MR | Zbl

[22] Keith S., Zhong X., “The Poincaré inequality is an open ended condition”, Ann. Math. (2), 167:2 (2008), 575–599 | DOI | MR | Zbl

[23] Skrypnik I. I., “Regulyarnost reshenii vyrozhdayuschikhsya kvazilineinykh parabolicheskikh uravnenii (vesovoi sluchai)”, Ukr. matem. zh., 48:7 (1996), 972–988 | MR | Zbl

[24] Surnachev M., “A Harnack inequality for weighted degenerate parabolic equations”, J. Diff. Eq., 248:8 (2010), 2092–2129 | DOI | MR | Zbl

[25] Vespri V., “On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations”, Manuscripta Math., 75:1 (1992), 65–80 | DOI | MR | Zbl

[26] Porzio M. M., Vespri V., “Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations”, J. Diff. Eq., 103:1 (1993), 146–178 | DOI | MR | Zbl

[27] Vespri V., “Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations”, J. Math. Anal. Appl., 181:1 (1994), 104–131 | DOI | MR | Zbl