YAO Feng-ping, ZHOU Shu-lin. Schauder Estimates for the Parabolic Equation of the Bi-Harmonic Type[J]. Applied Mathematics and Mechanics, 2007, 28(11): 1340-1352.
Citation: YAO Feng-ping, ZHOU Shu-lin. Schauder Estimates for the Parabolic Equation of the Bi-Harmonic Type[J]. Applied Mathematics and Mechanics, 2007, 28(11): 1340-1352.

Schauder Estimates for the Parabolic Equation of the Bi-Harmonic Type

  • Received Date: 2006-10-23
  • Rev Recd Date: 2007-08-20
  • Publish Date: 2007-11-15
  • Global Schauder estimates for the initial-value parabolic problem of the bi-harmonic type were proved. The existence and uniqueness of the solutions in the suitable space were obtained. Similarly to the second-order case, a fomal expression of solutions by the Fourier transform was obtained. Then the regularity, uniqueness, existence of solutions using the potential theory and approximation argument were got. The approach is simple and straightforward.
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  • [1]
    Schauder J.ber lineare elliptische Differentialgleichungen zweiter Ordnung[J].Math Z,1934,38(1):257-282. doi: 10.1007/BF01170635
    Schauder J.Numerische Abschdtzungen in elliptischen linearen Differentialgleichungen[J].Studia Math,1934,5(1):34-42.
    Campanato S.Propriet di una famiglia di spazi funzionali[J].Ann Scuola Norm Sup Pisa,1964,18(3):137-160.
    Trudinger N S.A new approach to the Schauder estimates for linear elliptic equations[J].Proc Centre Math Anal Austral Nat Univ,1986,14:52-59.
    Caffarelli L A.Interior a priori estimates for solutions of fully nonlinear equations[J].Ann Math,1989,130(1):189-213. doi: 10.2307/1971480
    Ciliberto C.Formule di maggiorazione e teoremi di esistenza per le soluzioni delle equazioni paraboliche in due variabili[J].Ricerche Mat,1954,3(1):40-75.
    Campanato S.Equazioni paraboliche del secondo ordine e spazi L[KG*2]. 2,λ(Ω;δ)[J].Ann Math Pura Appl,1966,73(4):55-102.
    Simon L.Schauder estimates by scaling[J].Calc Var PDE,1997,5(5):391-407. doi: 10.1007/s005260050072
    WANG Li-he.On the regularity theory of fully nonlinear parabolic equations Ⅱ[J].Comm Pure Appl Math,1992,45(2):141-178. doi: 10.1002/cpa.3160450202
    Friedman A.Partial Differential Equations of Parabolic Type[M].Englewood Cliffs,NJ:Prentice-Hall Inc,1964.
    Ladyzenskaja O A,Solonnikov V A,Uralceva N N.Linear and Quasilinear Equations of Parabolic Type[M].Providence, RI:American Mathematical Society,1968.
    Lorenzi L.Schauder estimates for degenerate elliptic and parabolic problems with unbounded coefficients in RN[J].Differential Internat Equations,2005,18(5):531-566.
    Lunardi A.Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in RN[J].Studia Math,1998,128(2):171-198.
    Lunardi A.Schauder estimates for a class of degenerateelliptic and parabolic operators with unbounded coefficients in RN[J].Annali della Scuola Normale Superiore Pisa,1997,24(4):133-164.
    Solonnikov V A.On boundary value problems for linear parabolic systems of differential equations of general form[J].Trudy Mat Inst Steklov,1965,83:3-163.
    Cahn J W,Hilliard J E.Free energy of nonuniform system I. Interfacial free energy[J].J Chem Phys,1958,28(2):258-367. doi: 10.1063/1.1744102
    Alikakos N D,Fusco G.Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: the motion of bubbles[J].Arch Rat Mech Anal,1998,141(1):1-61. doi: 10.1007/s002050050072
    Rossi R.On two classes of generalized viscous Cahn-Hilliard equations[J].Comm Pure Appl Anal,2005,4(2):405-430. doi: 10.3934/cpaa.2005.4.405
    Kwembe T A.Existence and uniqueness of global solutions for the parabolic equation of the bi-harmonic type[J].Nonlinear Anal,2001,47(2):1321-1332. doi: 10.1016/S0362-546X(01)00268-1
    XU Meng,ZHOU Shu-lin.Existence and uniqueness of weak solutions for a generalized thin film equation[J].Nonlinear Anal,2005,60(4):755-774. doi: 10.1016/j.na.2004.01.013
    YIN Jing-xue,LIU Chang-chun.Regularity of solutions of the Cahn-Hilliard equation with concentration dependent mobility[J].Nonlinear Anal,2001,45(5):543-554. doi: 10.1016/S0362-546X(99)00406-X
    DiBenedetto E.Partial Differential Equations[M].Boston-Basel-Berlin:Birkhuser,1995.
    CHEN Ya-zhe,WU Lan-cheng.Second Order Elliptic Partial Differential Equations and Elliptic Systems[M].Providence,RI:American Mathematical Society,1998.
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