# American Institute of Mathematical Sciences

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 and 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, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474. [2] L. Boccardo, Problemi differenziali ellittici e parabolici con dati misure, Boll. Un. Mat. Ital. A, 11 (1997), 439-461. [3] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040. [4] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655. doi: 10.1080/03605309208820857. [5] L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539-551. [6] V. Bögelein, F. Duzaar and U. Gianazza, Porous medium type equations with measure data and potential estimates, SIAM J. Math. Anal., 45 (2013), 3283-3330. doi: 10.1137/130925323. [7] V. Bögelein, F. Duzaar and U. Gianazza, Sharp boundedness and continuity results for the singular porous medium equation, (2014), preprint available from: https://www.mittag-leffler.se/preprints/files/IML-1314f-31.pdf. [8] V. Bögelein, F. Duzaar and P. Marcellini, Parabolic systems with $p,q$-growth: a variational approach, Arch. Ration. Mech. Anal., 210 (2013), 219-267. doi: 10.1007/s00205-013-0646-4. [9] V. Bögelein, T. Lukkari and C. Scheven, The obstacle problem for the porous medium equation, (2014), preprint available from: https://www.mittag-leffler.se/preprints/files/IML-1314f-33.pdf [10] B. E. Dahlberg and C. E. Kenig, Non-negative solutions to fast diffusions, Rev. Mat. Iberoamericana, 4 (1988), 11-29. doi: 10.4171/RMI/61. [11] P. Daskalopoulos and C. E. Kenig, Degenerate Diffusions, EMS Tracts in Mathematics, 1, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/033. [12] A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl., 170 (1996), 207-240. doi: 10.1007/BF01758989. [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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 97-126. doi: 10.1016/j.anihpc.2005.02.006. [14] A. Dall'Aglio and L. Orsina, Nonlinear parabolic equations with natural growth conditions and $L^1$ data, Nonlinear Anal., 27 (1996), 59-73. doi: 10.1016/0362-546X(94)00363-M. [15] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics. Springer, New York, 2012. [16] E. DiBenedetto, Degenerate Parabolic Equations, Springer Universitext, Springer, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [17] T. Kilpeläinen and J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 591-613. [18] T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793. [19] J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435. doi: 10.1007/s10231-005-0160-x. [20] J. Kinnunen and P. Lindqvist, Definition and properties of supersolutions to the porous medium equation, J. Reine Angew. Math., 618 (2008), 135-168. doi: 10.1515/CRELLE.2008.035. [21] T. Lukkari, The porous medium equation with measure data, J. Evol. Equ., 10 (2010), 711-729. doi: 10.1007/s00028-010-0067-x. [22] T. Lukkari, The fast diffusion equation with measure data, Nonlinear Differ. Equ. Appl., 19 (2011), 329-343. doi: 10.1007/s00030-011-0131-4. [23] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

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##### References:
 [1] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. doi: 10.1007/BF01176474. [2] L. Boccardo, Problemi differenziali ellittici e parabolici con dati misure, Boll. Un. Mat. Ital. A, 11 (1997), 439-461. [3] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040. [4] L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655. doi: 10.1080/03605309208820857. [5] L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539-551. [6] V. Bögelein, F. Duzaar and U. Gianazza, Porous medium type equations with measure data and potential estimates, SIAM J. Math. Anal., 45 (2013), 3283-3330. doi: 10.1137/130925323. [7] V. Bögelein, F. Duzaar and U. Gianazza, Sharp boundedness and continuity results for the singular porous medium equation, (2014), preprint available from: https://www.mittag-leffler.se/preprints/files/IML-1314f-31.pdf. [8] V. Bögelein, F. Duzaar and P. Marcellini, Parabolic systems with $p,q$-growth: a variational approach, Arch. Ration. Mech. Anal., 210 (2013), 219-267. doi: 10.1007/s00205-013-0646-4. [9] V. Bögelein, T. Lukkari and C. Scheven, The obstacle problem for the porous medium equation, (2014), preprint available from: https://www.mittag-leffler.se/preprints/files/IML-1314f-33.pdf [10] B. E. Dahlberg and C. E. Kenig, Non-negative solutions to fast diffusions, Rev. Mat. Iberoamericana, 4 (1988), 11-29. doi: 10.4171/RMI/61. [11] P. Daskalopoulos and C. E. Kenig, Degenerate Diffusions, EMS Tracts in Mathematics, 1, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/033. [12] A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the $H$-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl., 170 (1996), 207-240. doi: 10.1007/BF01758989. [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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 97-126. doi: 10.1016/j.anihpc.2005.02.006. [14] A. Dall'Aglio and L. Orsina, Nonlinear parabolic equations with natural growth conditions and $L^1$ data, Nonlinear Anal., 27 (1996), 59-73. doi: 10.1016/0362-546X(94)00363-M. [15] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics. Springer, New York, 2012. [16] E. DiBenedetto, Degenerate Parabolic Equations, Springer Universitext, Springer, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [17] T. Kilpeläinen and J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1992), 591-613. [18] T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793. [19] J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435. doi: 10.1007/s10231-005-0160-x. [20] J. Kinnunen and P. Lindqvist, Definition and properties of supersolutions to the porous medium equation, J. Reine Angew. Math., 618 (2008), 135-168. doi: 10.1515/CRELLE.2008.035. [21] T. Lukkari, The porous medium equation with measure data, J. Evol. Equ., 10 (2010), 711-729. doi: 10.1007/s00028-010-0067-x. [22] T. Lukkari, The fast diffusion equation with measure data, Nonlinear Differ. Equ. Appl., 19 (2011), 329-343. doi: 10.1007/s00030-011-0131-4. [23] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
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