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Local existence and uniqueness of solutions of degenerate parabolic equations. (English) Zbl 0738.35033

Author’s summary: We consider a class of degenerate parabolic equations on a bounded domain with mixed boundary conditions. These problems arise, for example, in the study of flow through porous media. Under appropriate hypotheses, we establish the existence of a nonnegative solution which is obtainable as a monotone limit of solutions of quasilinear parabolic equations. This construction is used to establish uniqueness, comparison, and \(L^ 1\) continuous dependence theorems, as well as some results on blow up of solutions in finite time.

MSC:

35K65 Degenerate parabolic equations
35Q35 PDEs in connection with fluid mechanics
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
76S05 Flows in porous media; filtration; seepage
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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References:

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