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

1. 

Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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., ().   Google Scholar

[6]

A. N. Dao, J. I. Díaz and P. Sauvy, Quenching phenomenon of singular parabolic problems with $L^{1}$ initial data,, to appear., ().   Google Scholar

[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.  Google Scholar

[8]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries,, Research Notes in Mathematics, (1985).   Google Scholar

[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.   Google Scholar

[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}., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[19]

B. Kawohl, Remarks on Quenching,, \emph{Doc. Math., 1 (1996), 199.   Google Scholar

[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.  Google Scholar

[21]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, Nauka, (1967).   Google Scholar

[22]

H. A. Levine, Quenching and beyond: a survey of recent results,, in \emph{Nonlinear mathematical problems in industry, (1992), 501.   Google Scholar

[23]

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Dunod, (1969).   Google Scholar

[24]

L. Nirenberg, An extended interpolation inequality,, \emph{Ann. Scuola Norm. Sup. Pisa}, 3 (1966), 733.   Google Scholar

[25]

D. Phillips, Existence of solutions of quenching problems,, \emph{Appl. Anal.}, 24 (1987), 253.  doi: 10.1080/00036818708839668.  Google Scholar

[26]

J. M. Rakotoson, Regularity of a very weak solution for parabolic equations and applications,, \emph{Advances in Differential Equations}, 16 (2011), 867.   Google Scholar

[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.   Google Scholar

[28]

M. Winkler, Instantaneous shrinking of the support in degenerate parabolic equations with strong absorption,, \emph{Adv. Differential Equations}, 9 (2004), 625.   Google Scholar

[29]

M. Winkler, A strongly degenerate diffusion equation with strong absorption,, \emph{Math. Nachrichten}, 277 (2004), 83.  doi: 10.1002/mana.200310221.  Google Scholar

[30]

M. Winkler, Nonuniqueness in the quenching problem,, \emph{Math. Ann.}, 339 (2007), 559.  doi: 10.1007/s00208-007-0123-1.  Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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., ().   Google Scholar

[6]

A. N. Dao, J. I. Díaz and P. Sauvy, Quenching phenomenon of singular parabolic problems with $L^{1}$ initial data,, to appear., ().   Google Scholar

[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.  Google Scholar

[8]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries,, Research Notes in Mathematics, (1985).   Google Scholar

[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.   Google Scholar

[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}., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[19]

B. Kawohl, Remarks on Quenching,, \emph{Doc. Math., 1 (1996), 199.   Google Scholar

[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.  Google Scholar

[21]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, Nauka, (1967).   Google Scholar

[22]

H. A. Levine, Quenching and beyond: a survey of recent results,, in \emph{Nonlinear mathematical problems in industry, (1992), 501.   Google Scholar

[23]

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Dunod, (1969).   Google Scholar

[24]

L. Nirenberg, An extended interpolation inequality,, \emph{Ann. Scuola Norm. Sup. Pisa}, 3 (1966), 733.   Google Scholar

[25]

D. Phillips, Existence of solutions of quenching problems,, \emph{Appl. Anal.}, 24 (1987), 253.  doi: 10.1080/00036818708839668.  Google Scholar

[26]

J. M. Rakotoson, Regularity of a very weak solution for parabolic equations and applications,, \emph{Advances in Differential Equations}, 16 (2011), 867.   Google Scholar

[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.   Google Scholar

[28]

M. Winkler, Instantaneous shrinking of the support in degenerate parabolic equations with strong absorption,, \emph{Adv. Differential Equations}, 9 (2004), 625.   Google Scholar

[29]

M. Winkler, A strongly degenerate diffusion equation with strong absorption,, \emph{Math. Nachrichten}, 277 (2004), 83.  doi: 10.1002/mana.200310221.  Google Scholar

[30]

M. Winkler, Nonuniqueness in the quenching problem,, \emph{Math. Ann.}, 339 (2007), 559.  doi: 10.1007/s00208-007-0123-1.  Google Scholar

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