August  2014, 7(4): 737-760. doi: 10.3934/dcdss.2014.7.737

$L^r_{ loc}-L^\infty_{ loc}$ estimates and expansion of positivity for a class of doubly non linear 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, Italy

3. 

Dipartimento di Matematica “U. Dini”, Università degli Studi di Firenze, viale Morgagni, 67/A, 50134, Firenze, Italy

Received  September 2013 Revised  November 2013 Published  February 2014

In this paper we show some properties regarding the local behaviour of local weak solutions to a class of doubly nonlinear singular parabolic equations.
Citation: Simona Fornaro, Maria Sosio, Vincenzo Vespri. $L^r_{ loc}-L^\infty_{ loc}$ estimates and expansion of positivity for a class of doubly non linear singular parabolic equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 737-760. doi: 10.3934/dcdss.2014.7.737
References:
[1]

M. Bonforte and G. Grillo, Super and ultracontractive bounds for doubly nonlinear evolution equations, Rev. Mat. Iberoamericana, 22 (2006), 111-129.

[2]

M. Bonforte, R. G. Iagar and J. L. Vázquez, Local smoothing effects, positivity, and Harnack inequalities for the fast $p$-Laplacian equation, Advances in Math., 224 (2010), 2151-2215. doi: 10.1016/j.aim.2010.01.023.

[3]

M. Bonforte and J. L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations, Advances in Math., 223 (2010), 529-578. doi: 10.1016/j.aim.2009.08.021.

[4]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[5]

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.

[6]

S. Fornaro and M. Sosio, Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations, Adv. Differential Equations, 13 (2008), 139-168.

[7]

S. Fornaro, M. Sosio and V. Vespri, Energy estimates and integral Harnack inequality for some doubly nonlinear singular parabolic equations, Contemporary Mathematics, 594 (2013), 179-199. doi: 10.1090/conm/594/11785.

[8]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t =\Delta u^m$ when $0< m <1$, Trans. Amer. Math. Soc., 291 (1985), 145-158. doi: 10.1090/S0002-9947-1985-0797051-0.

[9]

A. S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations, Russian Math. Surveys, 42 (1987), 169-222.

[10]

A. V. Ivanov, Regularity for doubly nonlinear parabolic equations, Journal of Mathematical Sciences, 83 (1997). doi: 10.1007/BF02398459.

[11]

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, Journal of Mathematical Sciences, 84 (1997).

[12]

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

[13]

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.

[14]

D. Stan and J. L. Vázquez, Asymptotic behaviour of the doubly nonlinear diffusion equation on bounded domains, Nonlinear Analysis TMA, 77 (2013), 1-32. doi: 10.1016/j.na.2012.08.011.

[15]

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.

show all references

References:
[1]

M. Bonforte and G. Grillo, Super and ultracontractive bounds for doubly nonlinear evolution equations, Rev. Mat. Iberoamericana, 22 (2006), 111-129.

[2]

M. Bonforte, R. G. Iagar and J. L. Vázquez, Local smoothing effects, positivity, and Harnack inequalities for the fast $p$-Laplacian equation, Advances in Math., 224 (2010), 2151-2215. doi: 10.1016/j.aim.2010.01.023.

[3]

M. Bonforte and J. L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations, Advances in Math., 223 (2010), 529-578. doi: 10.1016/j.aim.2009.08.021.

[4]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[5]

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.

[6]

S. Fornaro and M. Sosio, Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations, Adv. Differential Equations, 13 (2008), 139-168.

[7]

S. Fornaro, M. Sosio and V. Vespri, Energy estimates and integral Harnack inequality for some doubly nonlinear singular parabolic equations, Contemporary Mathematics, 594 (2013), 179-199. doi: 10.1090/conm/594/11785.

[8]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t =\Delta u^m$ when $0< m <1$, Trans. Amer. Math. Soc., 291 (1985), 145-158. doi: 10.1090/S0002-9947-1985-0797051-0.

[9]

A. S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations, Russian Math. Surveys, 42 (1987), 169-222.

[10]

A. V. Ivanov, Regularity for doubly nonlinear parabolic equations, Journal of Mathematical Sciences, 83 (1997). doi: 10.1007/BF02398459.

[11]

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, Journal of Mathematical Sciences, 84 (1997).

[12]

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

[13]

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.

[14]

D. Stan and J. L. Vázquez, Asymptotic behaviour of the doubly nonlinear diffusion equation on bounded domains, Nonlinear Analysis TMA, 77 (2013), 1-32. doi: 10.1016/j.na.2012.08.011.

[15]

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.

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