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 - A, 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. doi: 10.1016/0022-1236(77)90051-9.

[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. doi: 10.1137/0519070.

[3]

C. Caisheng, Global existence and $L^\infty$ estimates of solution for doubly nonlinear parabolic equation,, J. Math. Anal. Appl., 244 (2000), 133. doi: 10.1006/jmaa.1999.6695.

[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. doi: 10.1137/S0036139901385345.

[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. doi: 10.5565/PUBLMAT_40296_18.

[6]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations,, Springer Monographs in Mathematics, (2012). doi: 10.1007/978-1-4614-1584-8.

[7]

S. Fornaro and M. Sosio, Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations,, Adv. Differential Equations, 13 (2008), 139.

[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. doi: 10.1090/conm/594/11785.

[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. doi: 10.3934/dcdss.2014.7.737.

[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.

[11]

A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 135.

[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. doi: 10.1088/0022-3727/34/16/322.

[13]

T. Kuusi, J. Siljander and J. M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations,, Indiana Univ. Math. J., 61 (2012), 399. doi: 10.1512/iumj.2012.61.4513.

[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. doi: 10.1137/S0036141094270370.

[15]

A. V. Ivanov, Regularity for doubly nonlinear parabolic equations,, J. Math. Sci., 83 (1997), 22. doi: 10.1007/BF02398459.

[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. doi: 10.1007/BF02399936.

[17]

J. L. Lions, Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires,, Dunod, (1969).

[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. doi: 10.1006/jdeq.1993.1045.

[19]

M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption,, J. Math. Anal. Appl., 132 (1988), 187. doi: 10.1016/0022-247X(88)90053-4.

[20]

V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations,, Manuscripta Math., 75 (1992), 65. doi: 10.1007/BF02567072.

[21]

V. Vespri, Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations,, J. Math. Anal. Appl., 181 (1994), 104. doi: 10.1006/jmaa.1994.1008.

show all references

References:
[1]

A. Bamberger, Étude d'une équation doublement non linéaire,, J. Functional Analysis, 24 (1977), 148. doi: 10.1016/0022-1236(77)90051-9.

[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. doi: 10.1137/0519070.

[3]

C. Caisheng, Global existence and $L^\infty$ estimates of solution for doubly nonlinear parabolic equation,, J. Math. Anal. Appl., 244 (2000), 133. doi: 10.1006/jmaa.1999.6695.

[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. doi: 10.1137/S0036139901385345.

[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. doi: 10.5565/PUBLMAT_40296_18.

[6]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations,, Springer Monographs in Mathematics, (2012). doi: 10.1007/978-1-4614-1584-8.

[7]

S. Fornaro and M. Sosio, Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations,, Adv. Differential Equations, 13 (2008), 139.

[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. doi: 10.1090/conm/594/11785.

[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. doi: 10.3934/dcdss.2014.7.737.

[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.

[11]

A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 135.

[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. doi: 10.1088/0022-3727/34/16/322.

[13]

T. Kuusi, J. Siljander and J. M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations,, Indiana Univ. Math. J., 61 (2012), 399. doi: 10.1512/iumj.2012.61.4513.

[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. doi: 10.1137/S0036141094270370.

[15]

A. V. Ivanov, Regularity for doubly nonlinear parabolic equations,, J. Math. Sci., 83 (1997), 22. doi: 10.1007/BF02398459.

[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. doi: 10.1007/BF02399936.

[17]

J. L. Lions, Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires,, Dunod, (1969).

[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. doi: 10.1006/jdeq.1993.1045.

[19]

M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption,, J. Math. Anal. Appl., 132 (1988), 187. doi: 10.1016/0022-247X(88)90053-4.

[20]

V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations,, Manuscripta Math., 75 (1992), 65. doi: 10.1007/BF02567072.

[21]

V. Vespri, Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations,, J. Math. Anal. Appl., 181 (1994), 104. doi: 10.1006/jmaa.1994.1008.

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