Abstract
In this paper, we study the limit as m→ ∞ of changing sign solutions of the porous medium equation: u t = Δ:|u|m-1 u in a domain Ω of ℝN with Dirichlet boundary condition.
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References
Bénilan, PH., AStrong Regularity LP For Solutions of the Porous Media EquationIn C. Bardos et al., editorContribution to Nonlinear Pde.Pitman Research Notes, 1983.
Bénilan, PH.Quelques remarques sur la convergence singulière des semi-groupes linéairesPubl. Math. UFR Sci. Tech., Univ. Franche-Comté Besançon15(1995/97), 1–11.
Bénilan, PH., Boccardo, L. and Herrero, M.On the Limit of Solution of \( {u_t} = \Delta {u^m} as m \to \infty\), In M. Bertch et al., editor, in Some Topics in Nonlinear Pde’s, Torino, 1989. Proceedings Int.Conf.
Bénilan, PH. and Crandall, M. G.Regularizing effects of homogeneous evolution equationIn Baltimore, editor, Contributions to Analysis and Geometry. (1981), 23–30.
Bénilan, Pti. and Crandall, M. G.The Continuous Dependence on W of Solutions of \( {u_t} - \Delta \varphi \left( u \right) = 0\), Ind. Uni. Math. J. (30)2(1981), 162–177.
Bénilan, Ph., Crandall, M. G. And Sacks, P.Some L 1 Existence and Dependence Remit for Semilinear Elliptic Equation Under Nonlinear Boundary ConditionsAppl. Math. Optim. 17 (1988), 203–224.
Bénilan, PH. Evans, L. C. and Gariepy, R. F.On Some Singular Limits of Homogeneous SemigroupsJ. Evol. Equ. (the same volume).
Bénilan, PH. and Igbida, N.The mesa Problem for Neuman Boundary Value Problem(accepted in J. Differential Equations).
Bemlan, PH. and Igbida, N.Singular Limit for Perturbed Nonlinear SemigroupComm. Applied Nonlinear Anal. (4)3(1996), 23–42.
Bénilan, PH. and Igbida, N.Limite de la solution de \( {u_t} - \Delta {u^m} + div F\left( u \right) = 0 lorsque m \to \infty\), Rev. Mat. Complut. (13)1(2000), 195–205.
Brezis, H. and Pazy, A.Convergence And Approximation of Semigroups of Nonlinear Operators in Banach SpacesJ. Func. Anal. 9 (1972), 63–74.
Caffarelli, L. A. and Friedman, A.Asymptotic Behavior of Solution of \( {u_t} = \Delta {u^m} as m \to \infty\), Indiana Univ. Math. J. (1987), 711–728.
Dibenedetto, E. and Friedman, A.The ill-posed Hele-Shaw model and the Stefan problem for supercooled waterTrans. Amer. Math. Soc.282(1984), 183–204.
Elliot, C. M., Herrero, M. A., King, J. R. and Ockendon, J. R.The Mesa Patterns for \( {{u}_{t}} = \nabla \left( {{{u}^{m}}\nabla u} \right)as m \to \infty \). IMA J. Appl. Math.37(1986), 147–154.
Evans, L. C., Feldman, M. and Gariepy, R. F.Fast/slow diffusion and collapsing sandpilesJ. Differential Equations.137(1997), 166–209.
Friedman, A. and HOLLIG, K.On the Mesa ProblemJ. Math. Anal. Appl.123(1987), 564–571.
Friedman, A. and Huang, S.Asymptotic Behavior of Solutions of \( {u_t} = {\Delta _{\varphi m}}(u) as m \to \infty \) with Inconsistent Initial ValuesIn Analyse Mathématique et applications. Paris 1988.
Gil, O. and Quiros, F.Boundary layer formation in the transition from porous medium to a Hele-Shaw flowPreprint UAM.
Gil, O. and Quiros, F.Convergence of the porous media equation to Hele-ShawNonlinear Anal. TMA. (8)44(2001), 1111–1131.
Igbida, N.The mesa-limit of the porous medium equation and the Hele-Shaw problemDiff. and Integral Equations. (2)15(2002), 129–146.
Igbida, N.Limite Singulière de Problèmes d’Évolution Non LinéairesThèse de doctorat, Université de Franche-Comté, Juin 1997.
King, J. R., D. phil. thesis, Oxford University, 1986.
Lacey, A. A. Ockendon, J. R. and Tayler, J. B.“Waiting time” solutions of a nonlinear diffusion equationSIAM J. Appl. Math.6(1982), 1252–1264.
Quiros, F. and Vazquez, J. L.Asymptotic convergence of the Stefan problem to Hele-ShawTrans. Amer. Math. Soc. (2)353(2001), 609–634.
Zeldovich, Y. N. and Raizer, Y. P.Physics of shock waves and high temperature hydrodynamics phenomenaAcademic Press 1986.
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Igbida, N., Benilan, P. (2003). Singular limit of changing sign solutions of the porous medium equation. In: Arendt, W., Brézis, H., Pierre, M. (eds) Nonlinear Evolution Equations and Related Topics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7924-8_11
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DOI: https://doi.org/10.1007/978-3-0348-7924-8_11
Publisher Name: Birkhäuser, Basel
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