American Institute of Mathematical Sciences

H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102.  Google Scholar

H. Amann, Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function Spaces, Jindrich Necas Center for Mathematical Modeling Lecture Notes, 6, Matfyzpress, Prague, 2009.  Google Scholar

H. Amann, Linear and Quasilinear Parabolic Problems, Volume I, Abstract Linear Theory, Monographs in Mathematics, 89, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

S. N. Antontsev, C. R. Gonsalves and A. M. Meirmanov, Local existence of classical solutions to the well-posed Helle-Shaw problem, Port. Math. (N.S.), 59 (2002), 435-452.  Google Scholar

J.-H. Bailly, Local existence of classical solutions to first-order parabolic equations describing free boundaries, Nonlinear Anal., 32 (1998), 583-599. doi: 10.1016/S0362-546X(97)00504-X.  Google Scholar

B. V. Basaliy, I. I. Danilyuk and S. P. Degtyarev, Classical solvability of the multidimensional nonstationary filtration problem with free boundary (Russian. English summary), Dokl. Akad. Nauk Ukr. SSR, (1987), 3-7.  Google Scholar

B. V. Bazaliy and S. P. Degtyarev, On classical solvability of the multidimensional Stefan problem for convective motion of a viscous incompresssible fluid, Math. USSR Sb., 60 (1988), 1-17.  Google Scholar

B. V. Bazaliy and S. P. Degtyarev, Solvability of a problem with an unknown boundary between the domains of a parabolic and an elliptic equation, Ukr. Math. J., 41 (1989), 1155-1160. doi: 10.1007/BF01057253.  Google Scholar

B. V. Bazaliy and S. P. Degtyarev, Stefan problem with kinetic and classical conditions at the free boundary, Ukr. Math. J., 44 (1992), 139-148. doi: 10.1007/BF01061735.  Google Scholar

G. I. Bizhanova and V. A. Solonnikov, On problems with free boundaries for second-order parabolic equations, St. Petersburg Mathematical Journal, 12 (2001), 949-981.  Google Scholar

G. I. Bizhanova and V. A. Solonnikov, On some model problems for second order parabolic equations with time derivative in the boundary conditions, St. Petersbg. Math. J., 6 (1995), 1151-1166.  Google Scholar

Y.-K. Cho and D. Kim, A fourier multiplier theorem on the Besov-Lipschits spaces, Korean J. Math., 16 (2008), 85-90. Google Scholar

P. Constantin, D. Cyrdoba and F. Gancedo, On the global existence for the Muskat problem, J. Fur. Math. Soc. (JEMS), 15 (2013), 201-227. doi: 10.4171/JEMS/360.  Google Scholar

P. Constantin and M. Pugh, Global solutions for small data to the Hele-Shaw problem, Nonlinearity, 6 (1993), 393-415. doi: 10.1088/0951-7715/6/3/004.  Google Scholar

A. Cyrdoba, D. Cyrdoba and F. Gancedo, Porous media: The Muskat problem in three dimensions, Anal. PDE., 6 (1993), 447-497. doi: 10.2140/apde.2013.6.447.  Google Scholar

S. P. Degtyarev, Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder's estimates for a degenerate parabolic problem with dynamic boundary conditions, Nonlinear Differential Equations and Applications NoDEA, 22 (2015), 185-237. doi: 10.1007/s00030-014-0280-3.  Google Scholar

S. P. Degtyarev, The existence of a smooth interface in the evolutionary elliptic Muskat-Verigin problem with nonlinear source, (Russian) Ukrainian Mathematical Bulletin, 7 (2010), 301-330.  Google Scholar

S. P. Degtyarev, The existence of a smooth interface in the evolutionary elliptic Muskat-Verigin problem with nonlinear source,, , ().   Google Scholar

R. Denk, J. Prüss and R. Zacher, Maximal $L_p$ - regularity of parabolic problems with boundary dynamics of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187. doi: 10.1016/j.jfa.2008.07.012.  Google Scholar

R. Denk and R. Volevich, A new class of parabolic problems connected with Newton's polygon, Discrete Cont. Dyn. Syst., Dynamical Systems and Differential Equations, Proceedings of the 6th AIMS International Conference, suppl., (2007), 294-303.  Google Scholar

R. Denk and L. R. Volevich, Parabolic boundary value problems connected with Newton's polygon and some problems of crystallization, J. Evol. Equ., 8 (2008), 523-556. doi: 10.1007/s00028-008-0392-5.  Google Scholar

R. Denk and M. Kaip, General Parabolic Mixed Order Systems in $L_p$ and Applications, Operator Theory: Advances and Applications, 239, Birkhäuser/Springer, 2013. doi: 10.1007/978-3-319-02000-6.  Google Scholar

P. Dintelmann, Classes of Fourier multipliers and Besov-Nikolskij spaces, Math. Nachr., 173 (1995), 115-130. doi: 10.1002/mana.19951730108.  Google Scholar

H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Ration. Mech. Anal., 205 (2012), 119-149. doi: 10.1007/s00205-012-0501-z.  Google Scholar

H. Dong and S. Kim, Partial Scauder estimates for second-order elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 40 (2011), 481-500. doi: 10.1007/s00526-010-0348-9.  Google Scholar

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976.  Google Scholar

J. Escher and B.-V. Matioc, On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results, Z. Anal. Anwend., 30 (2011), 193-218. doi: 10.4171/ZAA/1431.  Google Scholar

J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 (1997), 1028-1047. doi: 10.1137/S0036141095291919.  Google Scholar

J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), 619-642.  Google Scholar

P. Fife, Schauder estimates under incomplete Hölder continuity assumptions, Pacific J. Math., 13 (1963), 511-550. doi: 10.2140/pjm.1963.13.511.  Google Scholar

A. Friedman, B. Hu and J. J. L. Velazquez, A Stefan problem for a protocell model with symmetry-breaking bifurcations of analitic solutions, Interfaces Free Bound., 3 (2001), 143-199. doi: 10.4171/IFB/37.  Google Scholar

A. Friedman and J. J. L. Velazquez, A free boundary problem associated with crystallization of polymers in a temperature field, Indiana Univ. Math. J., 50 (2001), 1609-1649. doi: 10.1512/iumj.2001.50.2118.  Google Scholar

E. Frolova, Solvability in Sobolev spaces of a problem for a second order parabolic equation with time derivative in the boundary condition, Portugal Math., 56 (1999), 419-441.  Google Scholar

C. G. Gal and H. Wu, Asymptotic behavior of Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst., 22 (2008), 1041-1063. doi: 10.3934/dcds.2008.22.1041.  Google Scholar

S. Gindikin and L. R. Volevich, The Method of Newton's Polyhedron in the Theory of Partial Differential Equations, Mathematics and its Applications, 86, Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-011-1802-6.  Google Scholar

M. Girardi and L. Weis, Operator-valued Fourier multiplier theorems on Besov spaces, Math. Nachr., 251 (2003), 34-51. doi: 10.1002/mana.200310029.  Google Scholar

G. R. Goldstein and A. Miranville, A Cahn-Hilliard-Gurtin model with dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. S., 6 (2013), 387-400. doi: 10.3934/dcdss.2013.6.387.  Google Scholar

K. K. Golovkin, On equivalent normalizations of fractional spaces, in Automatic Programming, Numerical Methods and Functional Analysis, Trudy Mat. Inst. Steklov., 66, Acad. Sci. USSR, Moscow-Leningrad, 1962, 364-383.  Google Scholar

K. K. Golovkin and V. A. Solonnikov, Bounds for integral operators in translation-invariant norms, in Boundary Value Problems of Mathematical Physics, Part 1, Collection of Articles, Trudy Mat. Inst. Steklov. Nauka, 70, Moscow-Leningrad, 1964, 47-58.  Google Scholar

K. K. Golovkin and V. A. Solonnikov, Estimates of integral operators in translation-invariant norms. II, in Boundary Value Problems of Mathematical Physics. Part 4, Trudy Mat. Inst. Steklov. 92, 1966, 5-30.  Google Scholar

K. K. Golovkin and V. A. Solonnikov, On some estimates of convolutions, in Boundary-Value Problems of Mathematical Physics and Related Problems of Function Theory. Part 2, Zap. Nauchn. Sem. LOMI, "Nauka", 7, Leningrad. Otdel., Leningrad, 1968, 6-86.  Google Scholar

B. Grec and E. V. Radkevich, Newton's polygon method and the local solvability of free boundary problems, Journal of Mathematical Sciences, 143 (2007), 3253-3292. doi: 10.1007/s10958-007-0208-0.  Google Scholar

D. Guidetti, The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are Hölder continuous with respect to space variables, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 7 (1996), 161-168.  Google Scholar

D. Guidetti, Optimal regularity for mixed parabolic problems in spaces of functions which are Hölder continuous with respect to space variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 763-790.  Google Scholar

V. N. Gusakov and S. P. Degtyarev, Existence of a smooth solution in a filtration problem, Ukr. Math. J., 41 (1989), 1027-1032. doi: 10.1007/BF01056273.  Google Scholar

L. Hörmander, The Analysis of Linear Partial Differential Operators. I, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

S. D. Ivasishen, Green's matrices of boundary value problems for Petrovskii parabolic systems of general form. I, Mathematics of the USSR-Sbornik, 42 (1982), 93-144. Google Scholar

S. D. Ivasishen, Green's matrices of boundary value problems for Petrovskii parabolic systems of general form. II, Mathematics of the USSR-Sbornik, 42 (1982), 461-498. Google Scholar

A. I. Komech, Linear partial differential equations with constant coefficients, in Partial Differential Equations II: Elements of the Modern Theory. Equations with Constant Coefficients, Encyclopaedia of Mathematical Sciences, 31, Springer, 2013, 121-255. doi: 10.1007/978-3-642-57876-2_2.  Google Scholar

J. Kovats, Real analytic solutions of parabolic equations with time-measurable coefficients, Proc. Amer. Math. Soc., 130 (2002), 1055-1064. doi: 10.1090/S0002-9939-01-06163-9.  Google Scholar

S. N. Kruzkov, A. Castro and M. M. Lopez, Schauder-type estimates and existence theorems for the solution of basic problems for linear and nonlinear parabolic equations, Sov. Math. Dokl., 16 (1975), 60-64. Google Scholar

N. Krylov, The Calderon-Zygmund theorem and parabolic equations in $L_p(\mathbbR,C^{2+\alpha})$- spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 1 (2002), 799-820.  Google Scholar

N. V. Krylov, Parabolic equations in $L_p$-spaces with mixed norms, St. Petersburg Mathematical Journal, 14 (2003), 603-614.  Google Scholar

N. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35 (2010), 1-22. doi: 10.1080/03605300903424700.  Google Scholar

Y. Kusaka and A. Tani, On the classical solvability of the two-phase Stefan problem in a viscous incompressible fluid flow, Math. Models Methods Appl. Sci., 12 (2002), 365-391. doi: 10.1142/S0218202502001696.  Google Scholar

Y. Kusaka, On a free boundary problem describing the phase transition in an incompressible viscous fluid, Interfaces Free Bound., 12 (2010), 157-185. doi: 10.4171/IFB/231.  Google Scholar

O. A. Ladyzhenskaya, A theorem on multiplicators in nonhomogeneous holder spaces and some of its applications, Journal of Mathematical Sciences (New York), 115 (2003), 2792-2802. doi: 10.1023/A:1023373920221.  Google Scholar

O. A. Ladyzhenskaya, On multiplicators in Hölder spaces with nonhomogeneous metric, Methods. Appl. Anal., 7 (2000), 465-472.  Google Scholar

G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity, Differential Integral Equations, 5 (1992), 1219-1236.  Google Scholar

L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM J. Math. Anal., 32 (2000), 588-615. doi: 10.1137/S0036141098342842.  Google Scholar

L. Lorenzi, Optimal Hölder regularity for nonautonomous Kolmogorov equations, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 169-191. doi: 10.3934/dcdss.2011.4.169.  Google Scholar

A. Lunardi, Maximal space regularity in nonhomogeneous initial-boundary value parabolic problem, Numer. Funct. Anal. Optim., 10 (1989), 323-349. doi: 10.1080/01630568908816306.  Google Scholar

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

A. M. Meirmanov, On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations, Math. USSR Sb., 40 (1981), 157-178. Google Scholar

A. Meirmanov, The Muskat problem for a viscoelastic filtration, Interfaces Free Bound., 13 (2011), 463-484. doi: 10.4171/IFB/268.  Google Scholar

I. Sh. Mogilevskiĭ and V. A. Solonnikov, Solvability of a noncoercive initial-boundary value problem for the Stokes system in Hölder classes of functions (the half-space case), Z. Anal. Anwendungen, 8 (1989), 329-347.  Google Scholar

J. Prüss, R. Racke and S. Zheng, Maximal regularity and a symptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions, Anali di Matematica, 185 (2006), 627-648. doi: 10.1007/s10231-005-0175-3.  Google Scholar

J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodinamically consistent Strfan problems, Arch. Ration. Mech. Anal., 207 (2013), 611-667. doi: 10.1007/s00205-012-0571-y.  Google Scholar

R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differential Equations, 8 (2003), 83-110.  Google Scholar

E. V. Radkevich, On conditions for the existence of a classical solution of the modified Stefan problem (Gibbs-Thomson low), Russ. Ac. Sc. Sb. Math., 75 (1993), 221-246. doi: 10.1070/SM1993v075n01ABEH003381.  Google Scholar

E. V. Radkevich, On the spectrum of the pencil in the Verigin-Muskat problem, Russ. Ac. Sc. Sb. Math., 80 (1995), 33-73. doi: 10.1070/SM1995v080n01ABEH003513.  Google Scholar

J. F. Rodrigues and V. A. Solonnikov, On a parabolic system with time derivative in the boundary conditions and related free boundary problems, Math. Ann., 315 (1999), 61-95. doi: 10.1007/s002080050318.  Google Scholar

E. Sinestrari, On the solutions of the first boundary value problem for the linear parabolic equations, Proc. Roy. Soc. Edinburgh, Sect.A., 108 (1988), 339-355. doi: 10.1017/S0308210500014712.  Google Scholar

V. A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations, in Boundary Value Problems of Mathematical Physics. Part 1, Collection of Articles, Trudy Mat. Inst. Steklov. Nauka, 70, Moscow-Leningrad, 1964, 213-317.  Google Scholar

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, in Proceedings of the Steklov Institute of Mathematics, 83, 1965, 3-163.  Google Scholar

V. A. Solonnikov, Estimates of solutions to some noncoercive initial-boundary value problems with the help of a theorem about multipliers in Laplace-Fourier integrals, (Russian) in Functional and Numerical Methods of Mathematical Physics, Naukova Dumka, Kyjiv, 1988, 220-228.  Google Scholar

V. A. Solonnikov, Estimates of solutions of the second initial-boundary problem for the Stokes system in the spaces of functions with Holder continuous derivatives with respect to spatial variables, Journal of Mathematical Sciences (New York), 109 (2002), 1997-2017. doi: 10.1023/A:1014456711451.  Google Scholar

G. Tian and X.-J. Wang, Partial regularity for elliptic equatios, Discrete Contin. Dyn. Syst., 28 (2010), 899-913. doi: 10.3934/dcds.2010.28.899.  Google Scholar

H. Triebel, Theory of Function Spaces II, Modern Birkhauser Classics, Birkhäuser, Basel, 2010.  Google Scholar

J. L. Vázquez and E. Vitillaro, On the Laplace equation with dynamical boundary conditions of reactive-diffusive type, J. Math. Anal. Appl., 354 (2009), 674-688. doi: 10.1016/j.jmaa.2009.01.023.  Google Scholar

J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Comm. Partial Differential Equations, 33 (2008), 561-612. doi: 10.1080/03605300801970960.  Google Scholar

H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with Wentzell boundary condition, Asymptot. Anal., 54 (2007), 71-92.  Google Scholar

D. Yang, W. Yuan and C. Zhuo, Musielak-Orlicz Besov-type and Triebel-Lizorkin-type spaces, Rev. Mat. Complut., 27 (2014), 93-157. doi: 10.1007/s13163-013-0120-8.  Google Scholar

F. Yi, Local classical solution of Muskat free boundary problem, J. Partial Differential Equations., 9 (1996), 84-96.  Google Scholar

F. Yi, Global classical solution of Muskat free boundary problem, J. Math. Anal. Appl., 288 (2003), 442-461. doi: 10.1016/j.jmaa.2003.09.003.  Google Scholar