# American Institute of Mathematical Sciences

December  2015, 35(12): 5909-5926. doi: 10.3934/dcds.2015.35.5909

## Harnack type inequalities for some doubly nonlinear singular parabolic equations

 1 Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia 2 Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, via Ferrata, 1, 27100, Pavia 3 Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, viale Morgagni, 67/A, 50134, Firenze, Italy

Received  March 2014 Published  May 2015

We prove Harnack type inequalities for a wide class of parabolic doubly nonlinear equations including $u_t=$ ${ div}(|u|^{m-1}|Du|^{p-2}Du)$. We will distinguish between the supercritical range $3 - \frac {p} {N} < p+m < 3$ and the subcritical $2 < p+m \le 3 - \frac {p} {N}$ range. Our results extend similar estimates holding for general equations having the same structure as the parabolic $p$-Laplace or the porous medium equation and recently collected in [6].
Citation: Simona Fornaro, Maria Sosio, Vincenzo Vespri. Harnack type inequalities for some doubly nonlinear singular parabolic equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5909-5926. doi: 10.3934/dcds.2015.35.5909
##### References:
 [1] A. Bamberger, Étude d'une équation doublement non linéaire, J. Functional Analysis, 24 (1977), 148-155. doi: 10.1016/0022-1236(77)90051-9.  Google Scholar [2] D. Blanchard and G. A. Francfort, Study of a doubly nonlinear heat equation with no growth assumptions on the parabolic term, SIAM J. Math. Anal., 19 (1988), 1032-1056. doi: 10.1137/0519070.  Google Scholar [3] C. Caisheng, Global existence and $L^\infty$ estimates of solution for doubly nonlinear parabolic equation, J. Math. Anal. Appl., 244 (2000), 133-146. doi: 10.1006/jmaa.1999.6695.  Google Scholar [4] N. Calvo, J. I. Díaz, J. Durany, E. Schiavi and C. Vázquez, On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics, SIAM J. Appl. Math., 63 (2002), 683-707 (electronic). doi: 10.1137/S0036139901385345.  Google Scholar [5] J. I. Díaz and J. F. Padial, Uniqueness and existence of solutions in the $BV_t(Q)$ space to a doubly nonlinear parabolic problem, Publ. Mat., 40 (1996), 527-560. doi: 10.5565/PUBLMAT_40296_18.  Google Scholar [6] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, Springer Verlag, New York, 2012. doi: 10.1007/978-1-4614-1584-8.  Google Scholar [7] S. Fornaro and M. Sosio, Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations, Adv. Differential Equations, 13 (2008), 139-168.  Google Scholar [8] S. Fornaro, M. Sosio and V. Vespri, Energy estimates and integral Harnack inequality for some doubly nonlinear singular parabolic equations, Contemp. Math., 594 (2013), 179-199. doi: 10.1090/conm/594/11785.  Google Scholar [9] S. Fornaro, M. Sosio and V. Vespri, $L_{loc}^r - L_{loc}^\infty$ estimates and expansion of positivity for a class of doubly non linear singular parabolic equations, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 737-760. doi: 10.3934/dcdss.2014.7.737.  Google Scholar [10] A. S. Kalashnikov, Propagation of perturbations in the first boundary value problem for a degenerate parabolic equation with a double nonlinearity, Trudy Sem. Petrovsk., (1982), {128-134}.  Google Scholar [11] A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Russian Math. Surveys, 42 (1987), 135-176, 287.  Google Scholar [12] M. Küntz and P. Lavallée, Experimental evidence and theoretical analysis of anomalous diffusion during water infiltration in porous building materials, J. Phys. D: Appl. Phys., 34 (2001), 2547-2554. doi: 10.1088/0022-3727/34/16/322.  Google Scholar [13] T. Kuusi, J. Siljander and J. M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations, Indiana Univ. Math. J., 61 (2012), 399-430. doi: 10.1512/iumj.2012.61.4513.  Google Scholar [14] K. Ishige, On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation, SIAM J. Math. Anal., 27 (1996), 1235-1260. doi: 10.1137/S0036141094270370.  Google Scholar [15] A. V. Ivanov, Regularity for doubly nonlinear parabolic equations, J. Math. Sci., 83 (1997), 22-37. doi: 10.1007/BF02398459.  Google Scholar [16] A. V. Ivanov, P. Z. Mkrtychan and W. Jäger, Existence and uniqueness of a regular solution of the Cauchy-Diriclhet problem for a class of doubly nonlinear parabolic equations, J. Math. Sci., 84 (1997), 845-855. doi: 10.1007/BF02399936.  Google Scholar [17] J. L. Lions, Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar [18] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Diff. Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.  Google Scholar [19] M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl., 132 (1988), {187-212}. doi: 10.1016/0022-247X(88)90053-4.  Google Scholar [20] V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations, Manuscripta Math., 75 (1992), {65-80}. doi: 10.1007/BF02567072.  Google Scholar [21] V. Vespri, Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations, J. Math. Anal. Appl., 181 (1994), 104-131. doi: 10.1006/jmaa.1994.1008.  Google Scholar

show all references

##### References:
 [1] A. Bamberger, Étude d'une équation doublement non linéaire, J. Functional Analysis, 24 (1977), 148-155. doi: 10.1016/0022-1236(77)90051-9.  Google Scholar [2] D. Blanchard and G. A. Francfort, Study of a doubly nonlinear heat equation with no growth assumptions on the parabolic term, SIAM J. Math. Anal., 19 (1988), 1032-1056. doi: 10.1137/0519070.  Google Scholar [3] C. Caisheng, Global existence and $L^\infty$ estimates of solution for doubly nonlinear parabolic equation, J. Math. Anal. Appl., 244 (2000), 133-146. doi: 10.1006/jmaa.1999.6695.  Google Scholar [4] N. Calvo, J. I. Díaz, J. Durany, E. Schiavi and C. Vázquez, On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics, SIAM J. Appl. Math., 63 (2002), 683-707 (electronic). doi: 10.1137/S0036139901385345.  Google Scholar [5] J. I. Díaz and J. F. Padial, Uniqueness and existence of solutions in the $BV_t(Q)$ space to a doubly nonlinear parabolic problem, Publ. Mat., 40 (1996), 527-560. doi: 10.5565/PUBLMAT_40296_18.  Google Scholar [6] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, Springer Verlag, New York, 2012. doi: 10.1007/978-1-4614-1584-8.  Google Scholar [7] S. Fornaro and M. Sosio, Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations, Adv. Differential Equations, 13 (2008), 139-168.  Google Scholar [8] S. Fornaro, M. Sosio and V. Vespri, Energy estimates and integral Harnack inequality for some doubly nonlinear singular parabolic equations, Contemp. Math., 594 (2013), 179-199. doi: 10.1090/conm/594/11785.  Google Scholar [9] S. Fornaro, M. Sosio and V. Vespri, $L_{loc}^r - L_{loc}^\infty$ estimates and expansion of positivity for a class of doubly non linear singular parabolic equations, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 737-760. doi: 10.3934/dcdss.2014.7.737.  Google Scholar [10] A. S. Kalashnikov, Propagation of perturbations in the first boundary value problem for a degenerate parabolic equation with a double nonlinearity, Trudy Sem. Petrovsk., (1982), {128-134}.  Google Scholar [11] A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Russian Math. Surveys, 42 (1987), 135-176, 287.  Google Scholar [12] M. Küntz and P. Lavallée, Experimental evidence and theoretical analysis of anomalous diffusion during water infiltration in porous building materials, J. Phys. D: Appl. Phys., 34 (2001), 2547-2554. doi: 10.1088/0022-3727/34/16/322.  Google Scholar [13] T. Kuusi, J. Siljander and J. M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations, Indiana Univ. Math. J., 61 (2012), 399-430. doi: 10.1512/iumj.2012.61.4513.  Google Scholar [14] K. Ishige, On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation, SIAM J. Math. Anal., 27 (1996), 1235-1260. doi: 10.1137/S0036141094270370.  Google Scholar [15] A. V. Ivanov, Regularity for doubly nonlinear parabolic equations, J. Math. Sci., 83 (1997), 22-37. doi: 10.1007/BF02398459.  Google Scholar [16] A. V. Ivanov, P. Z. Mkrtychan and W. Jäger, Existence and uniqueness of a regular solution of the Cauchy-Diriclhet problem for a class of doubly nonlinear parabolic equations, J. Math. Sci., 84 (1997), 845-855. doi: 10.1007/BF02399936.  Google Scholar [17] J. L. Lions, Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar [18] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Diff. Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.  Google Scholar [19] M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. Math. Anal. Appl., 132 (1988), {187-212}. doi: 10.1016/0022-247X(88)90053-4.  Google Scholar [20] V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations, Manuscripta Math., 75 (1992), {65-80}. doi: 10.1007/BF02567072.  Google Scholar [21] V. Vespri, Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations, J. Math. Anal. Appl., 181 (1994), 104-131. doi: 10.1006/jmaa.1994.1008.  Google Scholar
 [1] Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99. [2] Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605 [3] Liangjun Weng. The interior gradient estimate for some nonlinear curvature equations. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1601-1612. doi: 10.3934/cpaa.2019076 [4] Goro Akagi. Doubly nonlinear parabolic equations involving variable exponents. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 1-16. doi: 10.3934/dcdss.2014.7.1 [5] Alessandro Audrito. Bistable reaction equations with doubly nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 2977-3015. doi: 10.3934/dcds.2019124 [6] Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723 [7] A. Kh. Khanmamedov. Long-time behaviour of doubly nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373 [8] Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801 [9] Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5167-5176. doi: 10.3934/dcds.2013.33.5167 [10] Mitsuharu Ôtani, Yoshie Sugiyama. Lipschitz continuous solutions of some doubly nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 647-670. doi: 10.3934/dcds.2002.8.647 [11] Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052 [12] Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51 [13] Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153 [14] Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363 [15] W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431 [16] Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793 [17] Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975 [18] Goro Akagi. Doubly nonlinear evolution equations and Bean's critical-state model for type-II superconductivity. Conference Publications, 2005, 2005 (Special) : 30-39. doi: 10.3934/proc.2005.2005.30 [19] Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817 [20] Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15

2020 Impact Factor: 1.392