Abstract
By the use of Moser iteration and Campanato space estimate, theL ∞ andC α regularity estimate of the gradient of solutions of nonlinear parabolic system\(\frac{{\partial u^i }}{{\partial t}} - \nabla \cdot (g\left| {\nabla u} \right|)\nabla u^i ) = 0\) with nonstandard growth conditions are obtained under the natural structure constraints.
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References
Giorgi E De. Sulla differenziabilità e l'analiticità degli integrali mutipli regolari. Mem Accad Sci Torino CI Sci Fis Mat Natur 3, 1957: 3, 25–43
Ural'tseva N N. Degenerate quasilinear systems. Zap Naucn Sem Leninggrad Otdel Mat Inst Steklov, 1968, 7: 184–222
Uhlenback K. Regularity for a class of nonlinear elliptic systems Acta Math, 1977, 138: 219–240
Giaquinta M. Multiple integrals in the calculus variations and nonlinear elliptic systems. Annals of Mathematics Studies Princeton New Jersey, 1983
Benedetto E Di, Friedman A. Hölder estimate for nonlinear degenerate parabolic systems. J reineangew Math, 1985, 357: 1–22
Lieberman G M. Boundary regularity for solutions for degenerate elliptic equations. Non Anal TMA, 1988, 12: 1203–1219
Lin F, Yi L. BoundaryC 1,α regularity for variational inequalities. Comm Pure Appl Math, 1991, 44: 715–732
Lieberman G M. Boundary and initial regularity for solutions of degenerate parabolic equations. Non Anal TMA, 1993, 20: 551–569
Chen Y Z, Benedetto E Di. Boundary regularity for solutions of nonlinear degenerate parabolic systems. J reine angew Math, 1989, 395: 102–131
Choe H J. Hölder continuity for solutions of certain degenerate parabolic systems. Non Anal TMA, 1992, 18: 235–243
Donaldson T K. Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces. J Diff Equ, 1971, 10: 507–528
Donaldson E K, Trudinger N S. Orlicz-Sobolev spaces and imbedding theorem. J Func Anal, 1971, 8: 52–75
Donaldson T K. Orlicz-Sobolev spaces and nonlinear parabolic initial value problems. J Diff Equ, 1974, 16: 201–256
Marcellini P. Regularity and existence of solutions of elliptic equations withp, q-growth conditions. J Diff Equs, 1991, 90: 1–30
Lieberman G M. The natural generalization of natural conditions, of Ladyzhenskaya and Ural'tseva for elliptic equations. Comm P D E, 1991, 16: 311–361
Choe H J. Interior behavior of minimizers for certain functionals with nonstandard growth. Non Anal TMA, 1992, 19: 933–945
Benedetto E Di. On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann Sc Norm Sup Pisa Cp Sc Ser IV, 1986, 18: 487–535
Ladyzhenskaya O A, Solonnikov V A, Ural'tseva N N. Linear and Quasilinear Equations of Parabolic Type. Providence R I, 1968
Benedetto E Di, Friedman A. Regularity of solutions of nonlinear degenerate parabolic systems. J reine angew Math, 1984, 349: 83–128
Chen Y Z, Benedetto E Di. On the local behavior of solutions of singular parabolic equations. Arch Rat Mech Anal, 1988, 103: 319–345
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This work was supported by NNSF of China under Grant 39570223 and Grant 19675005
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Jiangsheng, Y. Regularity of solutions of certain parabolic system with nonstandard growth condition. Acta Mathematica Sinica 14, 145–160 (1998). https://doi.org/10.1007/BF02560201
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DOI: https://doi.org/10.1007/BF02560201
Keywords
- Nonstandard growth condition
- Degenerate and singular Equations
- Moser iteration
- Campanato space estimate