On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolev spaces. I, II. (English) Zbl 0659.35045

Math. Z. 203, No. 1, 153-171 (1990); No. 4, 569-579 (1990).
A nonlinear second order elliptic-parabolic system in a bounded domain with mixed Dirichlet-Neumann type boundary conditions is studied. A very wide scale of growths in nonlinearities (in time and space derivatives) is included. The obtained existence results extend results by H. W. Alt and S. Luckhaus who consider polynomial growth in the space derivatives. In a quasidiagonal case of the considered system (part II) the boundedness of the variational solution is proved also for a slightly degenerate elliptic part. A local existence result is obtained under the weak restrictions on reaction terms in the considered system. The method used is based on energy type a priori estimates, techniques of monotone operators in Orlicz-Sobolev spaces. In Part I a full discretization (in time and space) and in Part II a semidiscretization (in time) approximation is used.
Reviewer: J.Kačur


35J65 Nonlinear boundary value problems for linear elliptic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35B45 A priori estimates in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B35 Stability in context of PDEs
Full Text: DOI


[1] Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z.183, 311–341 (1983) · doi:10.1007/BF01176474
[2] Crandall, M.G.: Nonlinear semigroups and evolution governed by accretive operators. Proc. Symp. Pure Math.45, 305–337 (1986) · Zbl 0637.47039 · doi:10.1090/pspum/045.1/843569
[3] Doktor, P.: On the density of smooth functions in certain sub-spaces of Sobolev space. CMUC.14, 609–622 (1973) · Zbl 0268.46036
[4] Donaldson, T.: Inhomogeneous Orlicz-Sobolev spaces and nonlinear parabolic initial value problems. J. Differ. Equations16, 201–256 (1974) · Zbl 0286.35047 · doi:10.1016/0022-0396(74)90012-6
[5] Donaldson, T., Trudinger, N.S.: Orlicz-Sobolev spaces and imbedding theorems. J. Funct. Anal.8, 52–75 (1971) · Zbl 0216.15702 · doi:10.1016/0022-1236(71)90018-8
[6] Gossez, J.P.: Nonlinear elliptic boundary value problems for rapidly or slowly increasing coefficients. Trans. Am. Math. Soc.190, 163–205 (1974) · Zbl 0239.35045 · doi:10.1090/S0002-9947-1974-0342854-2
[7] Kačur, J.: Methiod of rothe in evolution equations. Teubner-Texte zur Mathematik. Leipzig80, (1985)
[8] Kačur J.: On existence of the weak solution for non-linar partial differential equations of elliptic type I, II. CMUC.11, 137–181 (1970); CMUC.13, 211–225 (1972) · Zbl 0195.39503
[9] Krasnosel’skiî, M., Rustickiî J.V.: Convex functions and Orlicz spaces. GITTL Moscow (1958)
[10] Robert, J.: Inequations variationnelles paraboliques fortement non linèaires. J. Math. Pures Appl.53, 299–321 (1974) · Zbl 0273.35047
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