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 - 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.  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.  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.  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.  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.  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, (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.   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.  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.  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.   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.   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.  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.  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.  doi: 10.1137/S0036141094270370.  Google Scholar

[15]

A. V. Ivanov, Regularity for doubly nonlinear parabolic equations,, J. Math. Sci., 83 (1997), 22.  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.  doi: 10.1007/BF02399936.  Google Scholar

[17]

J. L. Lions, Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires,, Dunod, (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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (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.   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.  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.  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.   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.   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.  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.  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.  doi: 10.1137/S0036141094270370.  Google Scholar

[15]

A. V. Ivanov, Regularity for doubly nonlinear parabolic equations,, J. Math. Sci., 83 (1997), 22.  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.  doi: 10.1007/BF02399936.  Google Scholar

[17]

J. L. Lions, Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires,, Dunod, (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.  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.  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.  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.  doi: 10.1006/jmaa.1994.1008.  Google Scholar

[1]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[2]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[3]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[4]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[5]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[6]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026

[7]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[8]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[9]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[10]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[11]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[12]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005

[13]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[14]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[15]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[16]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[17]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

[18]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[19]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[20]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]