• Previous Article
    Stokes and Oseen flow with Coriolis force in the exterior domain
  • DCDS-S Home
  • This Issue
  • Next Article
    Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping
June  2008, 1(2): 329-338. doi: 10.3934/dcdss.2008.1.329

Generalizations of logarithmic Sobolev inequalities

1. 

Universität Rostock, Institut für Mathematik, Universitätsplatz 1, 18051 Rostock, Germany

Received  September 2006 Revised  April 2007 Published  March 2008

We generalize logarithmic Sobolev inequalities to logarithmic Gagliardo-Nirenberg inequalities, and apply these inequalities to prove ultracontractivity of the semigroup generated by the doubly nonlinear $p$-Laplacian

$\dot{u}=\Delta_p u^m.$

Our proof does not use Moser iteration, but shows that the time-dependent Lebesgue norm $\||u(t)|\|_{r(t)}$ stays bounded for a variable exponent $r(t)$ blowing up in arbitrary short time.

Citation: Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329
[1]

Maria J. Esteban. Gagliardo-Nirenberg-Sobolev inequalities on planar graphs. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2101-2114. doi: 10.3934/cpaa.2022051

[2]

Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations and Control Theory, 2022, 11 (3) : 975-1000. doi: 10.3934/eect.2021033

[3]

Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure and Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475

[4]

Simona Fornaro, Maria Sosio, Vincenzo Vespri. Harnack type inequalities for some doubly nonlinear singular parabolic equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5909-5926. doi: 10.3934/dcds.2015.35.5909

[5]

Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801

[6]

Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046

[7]

C. Fabry, Raul Manásevich. Equations with a $p$-Laplacian and an asymmetric nonlinear term. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 545-557. doi: 10.3934/dcds.2001.7.545

[8]

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

[9]

Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920

[10]

B. Abdellaoui, I. Peral. On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure and Applied Analysis, 2003, 2 (4) : 539-566. doi: 10.3934/cpaa.2003.2.539

[11]

Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623

[12]

Kerstin Does. An evolution equation involving the normalized $P$-Laplacian. Communications on Pure and Applied Analysis, 2011, 10 (1) : 361-396. doi: 10.3934/cpaa.2011.10.361

[13]

Goro Akagi. Doubly nonlinear parabolic equations involving variable exponents. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 1-16. doi: 10.3934/dcdss.2014.7.1

[14]

Alessandro Audrito. Bistable reaction equations with doubly nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 2977-3015. doi: 10.3934/dcds.2019124

[15]

Angela A. Albanese, Elisabetta M. Mangino. Analytic semigroups and some degenerate evolution equations defined on domains with corners. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 595-615. doi: 10.3934/dcds.2015.35.595

[16]

Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure and Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1

[17]

Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254

[18]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075

[19]

Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173

[20]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (206)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]