American Institute of Mathematical Sciences

September  2014, 13(5): 1799-1814. doi: 10.3934/cpaa.2014.13.1799

On the free boundary for quenching type parabolic problems via local energy methods

Received  October 2013 Revised  January 2014 Published  June 2014

We extend some previous local energy method to the study the free boundary generated by the solutions of quenching type parabolic problems involving a negative power of the unknown in the equation.
Citation: Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799
References:
 [1] S. N. Antontsev, J. I. Díaz and S. Shmarev, The support shrinking properties for solutions of quasilinear parabolic equations with strong absorption terms,, \emph{Annales de la Facult\'e des Sciences de Toulouse}, IV (1995), 5. [2] S. N. Antontsev, J. I. Diaz and S. I. Shmarev, Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics,, Birkh\, (2001). doi: 10.1007/978-1-4612-0091-8. [3] F. Bernis, Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption,, \emph{Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, 104 (1986), 1. doi: 10.1017/S030821050001903X. [4] H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations,, in \emph{Contributions to Nonlinear Functional Analysis (ed. E. Zarantonello)}, (1971), 101. [5] A. N. Dao and J. I. Díaz, The existence of renormalized solutions with right-hand side data integrable with respect to the distance to the boundary, and application of the energy method for free boundary problems to some solutions out of the energy space,, to appear., (). [6] A. N. Dao, J. I. Díaz and P. Sauvy, Quenching phenomenon of singular parabolic problems with $L^{1}$ initial data,, to appear., (). [7] J. Dávila and M. Montenegro, Existence and asymptotic behavior for a singular parabolic equation,, \emph{Transactions of the AMS}, 357 (2004), 1801. doi: 10.1090/S0002-9947-04-03811-5. [8] J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries,, Research Notes in Mathematics, (1985). [9] J. I. Díaz, Estimates on the location of the free boundary for the obstacle and Stefan problems by means of some energy methods,, \emph{Georgian Mathematical Journal}, 15 (2008), 455. [10] J. I. Díaz and J. Hernández, Positive and free boundary solutions to some singular nonlinear elliptic problems with absorption: an overview and open problems,, to appear in \emph{Electronic J. Differential Equations}., (). [11] J. I. Díaz, J. Hernández and F. J. Mancebo, Branches of positive and free boundary solutions for some singular quasilinear elliptic problems,, \emph{J. Math. Anal. Appl.}, 352 (2009), 449. doi: 10.1016/j.jmaa.2008.07.073. [12] J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms,, \emph{Milan J. Maths}, 79 (2011), 233. doi: 10.1007/s00032-011-0151-x. [13] J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some singular second order semilinear elliptic problems,, in preparation., (). [14] J. I. Díaz, J. M. Morel and L. Oswald, An elliptic equation with singular nonlinearity,, \emph{Comm. in Partial Differential Equations}, 12 (1987), 1333. doi: 10.1080/03605308708820531. [15] J. I. Díaz and L. Veron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations,, \emph{Trans. of Am. Math. Soc}., 290 (1985), 787. doi: 10.2307/2000315. [16] J. Giacomoni, P. Sauvy and S. Shmarev, Complete quenching for a quasilinear parabolic equation,, \emph{J. Math. Anal. Appl.}, 410 (2014), 607. doi: 10.1016/j.jmaa.2013.08.051. [17] J. Hernández and F. Mancebo, Singular elliptic and parabolic equations,, in \emph{Handbook of Differential equations (ed. M. Chipot and P. Quittner)}, (2006), 317. [18] H. Kawarada, On solutions of initial-boundary problem for $u_t=u_{xx}+1/(1-u)$,, \emph{Publ. Res. Inst. Math. Sci}., 10 (): 729. [19] B. Kawohl, Remarks on Quenching,, \emph{Doc. Math., 1 (1996), 199. [20] B. Kawohl and R. Kersner, On degenerate diffusion with very strong absorption,, \emph{Mathematical Methods in the Applied Sciences, 15 (1992), 469. doi: 10.1002/mma.1670150703. [21] O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, Nauka, (1967). [22] H. A. Levine, Quenching and beyond: a survey of recent results,, in \emph{Nonlinear mathematical problems in industry, (1992), 501. [23] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Dunod, (1969). [24] L. Nirenberg, An extended interpolation inequality,, \emph{Ann. Scuola Norm. Sup. Pisa}, 3 (1966), 733. [25] D. Phillips, Existence of solutions of quenching problems,, \emph{Appl. Anal.}, 24 (1987), 253. doi: 10.1080/00036818708839668. [26] J. M. Rakotoson, Regularity of a very weak solution for parabolic equations and applications,, \emph{Advances in Differential Equations}, 16 (2011), 867. [27] M. I. Vishik, Systèmes d'équations aux dérivé es partielles fortement elliptiques quasi-linéaires sous forme divergente,, \emph{Troudi Mosk. Mat. Obv.}, 12 (1963), 125. [28] M. Winkler, Instantaneous shrinking of the support in degenerate parabolic equations with strong absorption,, \emph{Adv. Differential Equations}, 9 (2004), 625. [29] M. Winkler, A strongly degenerate diffusion equation with strong absorption,, \emph{Math. Nachrichten}, 277 (2004), 83. doi: 10.1002/mana.200310221. [30] M. Winkler, Nonuniqueness in the quenching problem,, \emph{Math. Ann.}, 339 (2007), 559. doi: 10.1007/s00208-007-0123-1.

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References:
 [1] S. N. Antontsev, J. I. Díaz and S. Shmarev, The support shrinking properties for solutions of quasilinear parabolic equations with strong absorption terms,, \emph{Annales de la Facult\'e des Sciences de Toulouse}, IV (1995), 5. [2] S. N. Antontsev, J. I. Diaz and S. I. Shmarev, Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics,, Birkh\, (2001). doi: 10.1007/978-1-4612-0091-8. [3] F. Bernis, Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption,, \emph{Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, 104 (1986), 1. doi: 10.1017/S030821050001903X. [4] H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations,, in \emph{Contributions to Nonlinear Functional Analysis (ed. E. Zarantonello)}, (1971), 101. [5] A. N. Dao and J. I. Díaz, The existence of renormalized solutions with right-hand side data integrable with respect to the distance to the boundary, and application of the energy method for free boundary problems to some solutions out of the energy space,, to appear., (). [6] A. N. Dao, J. I. Díaz and P. Sauvy, Quenching phenomenon of singular parabolic problems with $L^{1}$ initial data,, to appear., (). [7] J. Dávila and M. Montenegro, Existence and asymptotic behavior for a singular parabolic equation,, \emph{Transactions of the AMS}, 357 (2004), 1801. doi: 10.1090/S0002-9947-04-03811-5. [8] J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries,, Research Notes in Mathematics, (1985). [9] J. I. Díaz, Estimates on the location of the free boundary for the obstacle and Stefan problems by means of some energy methods,, \emph{Georgian Mathematical Journal}, 15 (2008), 455. [10] J. I. Díaz and J. Hernández, Positive and free boundary solutions to some singular nonlinear elliptic problems with absorption: an overview and open problems,, to appear in \emph{Electronic J. Differential Equations}., (). [11] J. I. Díaz, J. Hernández and F. J. Mancebo, Branches of positive and free boundary solutions for some singular quasilinear elliptic problems,, \emph{J. Math. Anal. Appl.}, 352 (2009), 449. doi: 10.1016/j.jmaa.2008.07.073. [12] J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms,, \emph{Milan J. Maths}, 79 (2011), 233. doi: 10.1007/s00032-011-0151-x. [13] J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some singular second order semilinear elliptic problems,, in preparation., (). [14] J. I. Díaz, J. M. Morel and L. Oswald, An elliptic equation with singular nonlinearity,, \emph{Comm. in Partial Differential Equations}, 12 (1987), 1333. doi: 10.1080/03605308708820531. [15] J. I. Díaz and L. Veron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations,, \emph{Trans. of Am. Math. Soc}., 290 (1985), 787. doi: 10.2307/2000315. [16] J. Giacomoni, P. Sauvy and S. Shmarev, Complete quenching for a quasilinear parabolic equation,, \emph{J. Math. Anal. Appl.}, 410 (2014), 607. doi: 10.1016/j.jmaa.2013.08.051. [17] J. Hernández and F. Mancebo, Singular elliptic and parabolic equations,, in \emph{Handbook of Differential equations (ed. M. Chipot and P. Quittner)}, (2006), 317. [18] H. Kawarada, On solutions of initial-boundary problem for $u_t=u_{xx}+1/(1-u)$,, \emph{Publ. Res. Inst. Math. Sci}., 10 (): 729. [19] B. Kawohl, Remarks on Quenching,, \emph{Doc. Math., 1 (1996), 199. [20] B. Kawohl and R. Kersner, On degenerate diffusion with very strong absorption,, \emph{Mathematical Methods in the Applied Sciences, 15 (1992), 469. doi: 10.1002/mma.1670150703. [21] O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, Nauka, (1967). [22] H. A. Levine, Quenching and beyond: a survey of recent results,, in \emph{Nonlinear mathematical problems in industry, (1992), 501. [23] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Dunod, (1969). [24] L. Nirenberg, An extended interpolation inequality,, \emph{Ann. Scuola Norm. Sup. Pisa}, 3 (1966), 733. [25] D. Phillips, Existence of solutions of quenching problems,, \emph{Appl. Anal.}, 24 (1987), 253. doi: 10.1080/00036818708839668. [26] J. M. Rakotoson, Regularity of a very weak solution for parabolic equations and applications,, \emph{Advances in Differential Equations}, 16 (2011), 867. [27] M. I. Vishik, Systèmes d'équations aux dérivé es partielles fortement elliptiques quasi-linéaires sous forme divergente,, \emph{Troudi Mosk. Mat. Obv.}, 12 (1963), 125. [28] M. Winkler, Instantaneous shrinking of the support in degenerate parabolic equations with strong absorption,, \emph{Adv. Differential Equations}, 9 (2004), 625. [29] M. Winkler, A strongly degenerate diffusion equation with strong absorption,, \emph{Math. Nachrichten}, 277 (2004), 83. doi: 10.1002/mana.200310221. [30] M. Winkler, Nonuniqueness in the quenching problem,, \emph{Math. Ann.}, 339 (2007), 559. doi: 10.1007/s00208-007-0123-1.
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