Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A global approach to fully nonlinear parabolic problems

Authors: Athanassios G. Kartsatos and Igor V. Skrypnik
Journal: Trans. Amer. Math. Soc. 352 (2000), 4603-4640
MSC (1991): Primary 35K55; Secondary 35K30, 35K35
Published electronically: June 13, 2000
MathSciNet review: 1694294
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the general initial-boundary value problem

(1)         $\displaystyle{\frac{\partial u}{\partial t}-F(x,t,u,\mathcal{D}^{1}u, \mathcal{D}^{2}u)=f(x,t),\quad (x,t)\in Q_{T}\equiv \Omega \times (0,T),}$
(2)         $\displaystyle{G(x,t,u,\mathcal{D}^{1}u)=g(x,t),\quad (x,t)\in S_{T}\equiv \partial\Omega \times (0,T),}$
(3)         $\displaystyle{u(x,0)=h(x),\quad x\in \Omega,}$
where $\Omega $ is a bounded open set in $\mathcal{R}^{n}$ with sufficiently smooth boundary.  The problem (1)-(3) is first reduced to the analogous problem in the space $W^{(4),0}_{p}(Q_{T})$with zero initial condition and

\begin{displaymath}f\in W^{(2),0}_{p}(Q_{T}),~g \in W^{(3-\frac{1}{p}),0}_{p}(S_{T}). \end{displaymath}

The resulting problem is then reduced to the problem $Au=0,$ where the operator $A:W^{(4),0}_{p}(Q_{T})\to \left [W^{(4),0}_{p}(Q_{T})\right ]^{*}$ satisfies Condition $(S)_{+}.$  This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces.  The local and global solvability of the operator equation $Au=0$ are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.

References [Enhancements On Off] (What's this?)

  • [1] P. Acquistapace and B. Terreni, Fully nonlinear parabolic systems, Recent advances in nonlinear elliptic and parabolic problems (Nancy, 1988), Pitman Res. Notes Math. Ser., vol. 208, Longman Sci. Tech., Harlow, 1989, pp. 97–111. MR 1035000
  • [2] Herbert Amann, Quasilinear parabolic systems under nonlinear boundary conditions, Arch. Rational Mech. Anal. 92 (1986), no. 2, 153–192. MR 816618, 10.1007/BF00251255
  • [3] Guang Chang Dong, Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations, J. Partial Differential Equations Ser. A 1 (1988), no. 2, 12–42. MR 985445
  • [4] S. I. Hudjaev, The first boundary-value problem for non-linear parabolic equations, Dokl. Akad. Nauk SSSR 149 (1963), 535–538 (Russian). MR 0158178
  • [5] V. P. Il′in, Properties of certain classes of differentiable functions of several variables defined in an 𝑛-dimensional domain, Trudy Mat. Inst. Steklov 66 (1962), 227–363 (Russian). MR 0153789
  • [6] S. N. Kružkov, A. Kastro, and M. Lopes, Schauder type estimates, and theorems on the existence of the solution of fundamental problems for linear and nonlinear parabolic equations, Dokl. Akad. Nauk SSSR 220 (1975), 277–280 (Russian). MR 0393848
  • [7] N. V. Krylov, Nelineinye ellipticheskie i parabolicheskie uravneniya vtorogo poryadka, “Nauka”, Moscow, 1985 (Russian). MR 815513
    N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR 901759
  • [8] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
    O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). MR 0241822
  • [9] Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184
  • [10] A. Lunardi, Maximal space regularity in nonhomogeneous initial-boundary value parabolic problem, Numer. Funct. Anal. Optim. 10 (1989), no. 3-4, 323–349. MR 989538, 10.1080/01630568908816306
  • [11] Alessandra Lunardi, On a class of fully nonlinear parabolic equations, Comm. Partial Differential Equations 16 (1991), no. 1, 145–172. MR 1096836, 10.1080/03605309108820754
  • [12] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115–162. MR 0109940
  • [13] Kazuaki Taira, On a degenerate oblique derivative problem of Ju. V. Egorov and V. A. Kondrat′ev (Mat. Sb. (N.S.) 78(120) (1969), 148–176), J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 2, 383–391. MR 0435585
  • [14] I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, Translations of Mathematical Monographs, vol. 139, American Mathematical Society, Providence, RI, 1994. Translated from the 1990 Russian original by Dan D. Pascali. MR 1297765
  • [15] V. A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type, Trudy Mat. Inst. Steklov. 70 (1964), 133–212 (Russian). MR 0162065
  • [16] N. N. Šopolov, The first boundary value problem for nonlinear parabolic equations of arbitrary order, C. R. Acad. Bulgare Sci. 23 (1970), 899–902 (Russian). MR 0364876
  • [17] N. N. Ural′tseva, A nonlinear problem with an oblique derivative for parabolic equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 188 (1991), no. Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsii. 22, 143–158, 188 (Russian, with English summary); English transl., J. Math. Sci. 70 (1994), no. 3, 1817–1827. MR 1111473, 10.1007/BF02149151
  • [18] Lihe Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math. 45 (1992), no. 1, 27–76. MR 1135923, 10.1002/cpa.3160450103
    Lihe Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math. 45 (1992), no. 2, 141–178. MR 1139064, 10.1002/cpa.3160450202

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35K55, 35K30, 35K35

Retrieve articles in all journals with MSC (1991): 35K55, 35K30, 35K35

Additional Information

Athanassios G. Kartsatos
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

Igor V. Skrypnik
Affiliation: Institute for Applied Mathematics and Mechanics, R. Luxemburg Str. 74, Donetsk 340114, Ukraine

Keywords: Initial-boundary value problem, mapping of type $(S)_{+},$ Skrypnik's degree theory for demicontinuous mappings of type $(S)_{+},$ Galerkin approximation
Received by editor(s): April 18, 1997
Received by editor(s) in revised form: May 7, 1998
Published electronically: June 13, 2000
Additional Notes: This research was partially supported by an NSF-NRC COBASE grant.
Article copyright: © Copyright 2000 American Mathematical Society