Hölder continuity of weak solutions to nondiagonal degenerate parabolic system of three equations

D.Portnyagin

Hölder continuity of weak solutions is studied for a nondiagonal parabolic system of degenerate quasilinear differential equations with matrix of coefficients satisfying special structure conditions. A technique based on estimating the linear combinations of unknowns is employed to this end.

1. Amann H. Dymnamic theory of quasilinear parabolic equations, I Abstract evolution equations, Nonlinear Anal. 12 (1988), 895--919.

2. Amann H. Dymnamic theory of quasilinear parabolic systems, III Global existence, Math. Z. 202 (1989), 219--250.

3. Amann H. Dymnamic theory of quasilinear parabolic equations, II Reaction-diffusion systems, Differ. Integral Equations. 3 (1990), 13--75.

4. Bitsadze A. V. On elliptic systems of differential equations with partial derivatives of second order, Dokl. AN SSSR 112 (1957) no. 6, 983--986.

5. Bitsadze A. V. Boundary Value Problems for Second Order Elliptic Equations, Moskva, Nauka 1966.

6. Bitsadze A. V. Some Classes of Partial Differential Equations, Moskva, Nauka 1981.

7. Y. Z. Chen and L. C. Wu Second Order Elliptic Equations and Elliptic Systems Amer. Math. Soc. Providence, RI (1998).

8. DiBenedetto E. Degenerate Parabolic Equations, Springer, New York (1993).

9. Dung L. Hölder regularity for certain strongly coupled parabolic systems, J. Differential Equations 151 (1999), 313--344.

10. De Giorgi E. Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. (1968), 135--137.

11. Küfner K. H. W. Global existence for a certain strongly coupled quasilinear parabolic systems in population dynamics, Analysis 15 (1995), 343--357.

12. Ladyzhenskaya O. A., Solonnikov N. A., Ural'tseva N. N. Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Providence, RI (1968).

13. Portnyagin D. A generalization of the maximum principle to nonlinear parabolic systems, Annales Pol. Math. 81.3 (2003), 217--236.

14. Pozio M. A., Tesei A. Global existence of solutions for a strongly coupled quasilinear parabolic system, Nonlinear Anal. 12 (1990) No. 8, 657--689.

15. Wiegner M. Global solutions to a class of strongly coupled parabolic systems, Math. Ann. 292 (1992), 711--727.

	Hölder continuity of weak solutions to nondiagonal degenerate parabolic system of three equations
Author	D.Portnyagin Institute for Condensed Matter Physics of the National Academy of,Sciences of Ukraine
Abstract	Hölder continuity of weak solutions is studied for a nondiagonal parabolic system of degenerate quasilinear differential equations with matrix of coefficients satisfying special structure conditions. A technique based on estimating the linear combinations of unknowns is employed to this end.
Keywords	weak solution, Hölder continuity, degenerate parabolic system
DOI	doi:10.30970/ms.26.2.174-201
Reference	1. Amann H. Dymnamic theory of quasilinear parabolic equations, I Abstract evolution equations, Nonlinear Anal. 12 (1988), 895--919. 2. Amann H. Dymnamic theory of quasilinear parabolic systems, III Global existence, Math. Z. 202 (1989), 219--250. 3. Amann H. Dymnamic theory of quasilinear parabolic equations, II Reaction-diffusion systems, Differ. Integral Equations. 3 (1990), 13--75. 4. Bitsadze A. V. On elliptic systems of differential equations with partial derivatives of second order, Dokl. AN SSSR 112 (1957) no. 6, 983--986. 5. Bitsadze A. V. Boundary Value Problems for Second Order Elliptic Equations, Moskva, Nauka 1966. 6. Bitsadze A. V. Some Classes of Partial Differential Equations, Moskva, Nauka 1981. 7. Y. Z. Chen and L. C. Wu Second Order Elliptic Equations and Elliptic Systems Amer. Math. Soc. Providence, RI (1998). 8. DiBenedetto E. Degenerate Parabolic Equations, Springer, New York (1993). 9. Dung L. Hölder regularity for certain strongly coupled parabolic systems, J. Differential Equations 151 (1999), 313--344. 10. De Giorgi E. Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. (1968), 135--137. 11. Küfner K. H. W. Global existence for a certain strongly coupled quasilinear parabolic systems in population dynamics, Analysis 15 (1995), 343--357. 12. Ladyzhenskaya O. A., Solonnikov N. A., Ural'tseva N. N. Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Providence, RI (1968). 13. Portnyagin D. A generalization of the maximum principle to nonlinear parabolic systems, Annales Pol. Math. 81.3 (2003), 217--236. 14. Pozio M. A., Tesei A. Global existence of solutions for a strongly coupled quasilinear parabolic system, Nonlinear Anal. 12 (1990) No. 8, 657--689. 15. Wiegner M. Global solutions to a class of strongly coupled parabolic systems, Math. Ann. 292 (1992), 711--727.
Pages	174-201
Volume	26
Issue	2
Year	2006
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html