January  2015, 14(1): 23-49. doi: 10.3934/cpaa.2015.14.23

Very weak solutions of singular porous medium equations with measure data

1. 

Department Mathematik, Universität Erlangen--Nürnberg, Cauerstr. 11, 91056 Erlangen, Germany, Germany

2. 

Dipartimento di Matematica "F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia

Received  April 2014 Revised  April 2014 Published  September 2014

We consider non-homogeneous, singular ($ 0 < m < 1 $) porous medium type equations with a non-negative Radon-measure $\mu$ having finite total mass $\mu(E_T)$ on the right-hand side. We deal with a Cauchy-Dirichlet problem for these type of equations, with homogeneous boundary conditions on the parabolic boundary of the domain $E_T$, and we establish the existence of a solution in the sense of distributions. Finally, we show that the constructed solution satisfies linear pointwise estimates via linear Riesz potentials.
Citation: Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23
References:
[1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, \emph{Math. Z.}, 183 (1983), 311. doi: 10.1007/BF01176474. Google Scholar

[2]

L. Boccardo, Problemi differenziali ellittici e parabolici con dati misure,, \emph{Boll. Un. Mat. Ital. A}, 11 (1997), 439. Google Scholar

[3]

L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data,, \emph{J. Funct. Anal.}, 147 (1997), 237. doi: 10.1006/jfan.1996.3040. Google Scholar

[4]

L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures,, \emph{Comm. Partial Differential Equations}, 17 (1992), 641. doi: 10.1080/03605309208820857. Google Scholar

[5]

L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 13 (1996), 539. Google Scholar

[6]

V. Bögelein, F. Duzaar and U. Gianazza, Porous medium type equations with measure data and potential estimates,, \emph{SIAM J. Math. Anal.}, 45 (2013), 3283. doi: 10.1137/130925323. Google Scholar

[7]

V. Bögelein, F. Duzaar and U. Gianazza, Sharp boundedness and continuity results for the singular porous medium equation,, (2014), (2014). Google Scholar

[8]

V. Bögelein, F. Duzaar and P. Marcellini, Parabolic systems with $p,q$-growth: a variational approach,, \emph{Arch. Ration. Mech. Anal.}, 210 (2013), 219. doi: 10.1007/s00205-013-0646-4. Google Scholar

[9]

V. Bögelein, T. Lukkari and C. Scheven, The obstacle problem for the porous medium equation,, (2014), (2014). Google Scholar

[10]

B. E. Dahlberg and C. E. Kenig, Non-negative solutions to fast diffusions,, \emph{Rev. Mat. Iberoamericana}, 4 (1988), 11. doi: 10.4171/RMI/61. Google Scholar

[11]

P. Daskalopoulos and C. E. Kenig, Degenerate Diffusions,, EMS Tracts in Mathematics, (2007). doi: 10.4171/033. Google Scholar

[12]

A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations,, \emph{Ann. Mat. Pura Appl., 170 (1996), 207. doi: 10.1007/BF01758989. Google Scholar

[13]

A. Dall'Aglio, D. Giachetti, C. Leone and S. Segura de León, Quasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 23 (2006), 97. doi: 10.1016/j.anihpc.2005.02.006. Google Scholar

[14]

A. Dall'Aglio and L. Orsina, Nonlinear parabolic equations with natural growth conditions and $L^1$ data,, \emph{Nonlinear Anal.}, 27 (1996), 59. doi: 10.1016/0362-546X(94)00363-M. Google Scholar

[15]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations,, Springer Monographs in Mathematics. Springer, (2012). Google Scholar

[16]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer Universitext, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar

[17]

T. Kilpeläinen and J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 591. Google Scholar

[18]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, \emph{Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793. Google Scholar

[19]

J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation,, \emph{Ann. Mat. Pura Appl., 185 (2006), 411. doi: 10.1007/s10231-005-0160-x. Google Scholar

[20]

J. Kinnunen and P. Lindqvist, Definition and properties of supersolutions to the porous medium equation,, \emph{J. Reine Angew. Math., 618 (2008), 135. doi: 10.1515/CRELLE.2008.035. Google Scholar

[21]

T. Lukkari, The porous medium equation with measure data,, \emph{J. Evol. Equ., 10 (2010), 711. doi: 10.1007/s00028-010-0067-x. Google Scholar

[22]

T. Lukkari, The fast diffusion equation with measure data,, \emph{Nonlinear Differ. Equ. Appl., 19 (2011), 329. doi: 10.1007/s00030-011-0131-4. Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Ann. Mat. Pura Appl.}, 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

show all references

References:
[1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, \emph{Math. Z.}, 183 (1983), 311. doi: 10.1007/BF01176474. Google Scholar

[2]

L. Boccardo, Problemi differenziali ellittici e parabolici con dati misure,, \emph{Boll. Un. Mat. Ital. A}, 11 (1997), 439. Google Scholar

[3]

L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data,, \emph{J. Funct. Anal.}, 147 (1997), 237. doi: 10.1006/jfan.1996.3040. Google Scholar

[4]

L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures,, \emph{Comm. Partial Differential Equations}, 17 (1992), 641. doi: 10.1080/03605309208820857. Google Scholar

[5]

L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 13 (1996), 539. Google Scholar

[6]

V. Bögelein, F. Duzaar and U. Gianazza, Porous medium type equations with measure data and potential estimates,, \emph{SIAM J. Math. Anal.}, 45 (2013), 3283. doi: 10.1137/130925323. Google Scholar

[7]

V. Bögelein, F. Duzaar and U. Gianazza, Sharp boundedness and continuity results for the singular porous medium equation,, (2014), (2014). Google Scholar

[8]

V. Bögelein, F. Duzaar and P. Marcellini, Parabolic systems with $p,q$-growth: a variational approach,, \emph{Arch. Ration. Mech. Anal.}, 210 (2013), 219. doi: 10.1007/s00205-013-0646-4. Google Scholar

[9]

V. Bögelein, T. Lukkari and C. Scheven, The obstacle problem for the porous medium equation,, (2014), (2014). Google Scholar

[10]

B. E. Dahlberg and C. E. Kenig, Non-negative solutions to fast diffusions,, \emph{Rev. Mat. Iberoamericana}, 4 (1988), 11. doi: 10.4171/RMI/61. Google Scholar

[11]

P. Daskalopoulos and C. E. Kenig, Degenerate Diffusions,, EMS Tracts in Mathematics, (2007). doi: 10.4171/033. Google Scholar

[12]

A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations,, \emph{Ann. Mat. Pura Appl., 170 (1996), 207. doi: 10.1007/BF01758989. Google Scholar

[13]

A. Dall'Aglio, D. Giachetti, C. Leone and S. Segura de León, Quasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 23 (2006), 97. doi: 10.1016/j.anihpc.2005.02.006. Google Scholar

[14]

A. Dall'Aglio and L. Orsina, Nonlinear parabolic equations with natural growth conditions and $L^1$ data,, \emph{Nonlinear Anal.}, 27 (1996), 59. doi: 10.1016/0362-546X(94)00363-M. Google Scholar

[15]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations,, Springer Monographs in Mathematics. Springer, (2012). Google Scholar

[16]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer Universitext, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar

[17]

T. Kilpeläinen and J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 591. Google Scholar

[18]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, \emph{Acta Math., 172 (1994), 137. doi: 10.1007/BF02392793. Google Scholar

[19]

J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation,, \emph{Ann. Mat. Pura Appl., 185 (2006), 411. doi: 10.1007/s10231-005-0160-x. Google Scholar

[20]

J. Kinnunen and P. Lindqvist, Definition and properties of supersolutions to the porous medium equation,, \emph{J. Reine Angew. Math., 618 (2008), 135. doi: 10.1515/CRELLE.2008.035. Google Scholar

[21]

T. Lukkari, The porous medium equation with measure data,, \emph{J. Evol. Equ., 10 (2010), 711. doi: 10.1007/s00028-010-0067-x. Google Scholar

[22]

T. Lukkari, The fast diffusion equation with measure data,, \emph{Nonlinear Differ. Equ. Appl., 19 (2011), 329. doi: 10.1007/s00030-011-0131-4. Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Ann. Mat. Pura Appl.}, 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

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