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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

MSC:

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
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