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.


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
Full Text: DOI


[1] Adams R., Arch. Rat. Mech. Anal 92 pp 153– (1986)
[2] Ph. D. dissertation (1975)
[3] J. R. Anderson Stability and instability for solutions of the convective porous medium equation with at the boundary I. In Preparation
[4] Anderson J. R., T.S.R. Math Research center
[5] Bear J., Elsevier (1972)
[6] Bear J., E, in: NATO Asi Applied Sciences
[7] Benilan, P. 1981. ”Evolation Equations and Accretive Operators”. UNivresi9ty of Kentucky.
[8] R. S. Cantrell, C. Cosner Diffusive logistic equaitons with indefinite weights Populations models in disrupted environments II. Preprint
[9] Cantrell R. S., Proceedings od Symposia in Pure Matheatics (1986)
[10] DOI: 10.1016/0022-0396(87)90125-2 · Zbl 0634.35042
[11] Dibenedetto E., T.S.R. Math Research Center
[12] Friedman A., Prentice-Hall (1965)
[13] DOI: 10.1007/BF00280441 · Zbl 0366.76074
[14] Gilding B. H., Ann. Scuola Norm. sup. Pisa 4 pp 393– (1977)
[15] Gilding B. H., An Introduction to Variational Inequaitons and Their Applciation (1986)
[16] DOI: 10.1016/0022-0396(74)90018-7 · Zbl 0285.35035
[17] DOI: 10.1137/0519023 · Zbl 0696.35159
[18] DOI: 10.1137/0519023 · Zbl 0696.35159
[19] DOI: 10.1016/0022-247X(86)90314-8 · Zbl 0609.35021
[20] DOI: 10.1007/BF01774284 · Zbl 0658.35050
[21] Peletier L. A., The porous medium equaiton, in Application of Nonlinear Analysis in the Physical Sciences 148 (1981)
[22] Snacks P. E., T.S.R. Math Research Center (1968)
[23] DOI: 10.1016/0362-546X(83)90092-5 · Zbl 0511.35052
[24] Wolanski N. I., Trabajos de Matematica Publicacioned Previas Instituto Argentino de Mathematica 7 pp 1– (1984)
[25] DOI: 10.1016/0022-247X(85)90182-9 · Zbl 0595.76097
[26] xu X., Preprint 109 (1985)
[27] Walter W., Differential and Integral Inequalities (1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.