American Institute of Mathematical Sciences

Advanced
April  2002, 8(2): 451-468. doi: 10.3934/dcds.2002.8.451

Nonlinear parabolic differential equations and inequalities

Wolfgang Walter 1,

1. 

Math. Institut I, Universität Karlsruhe, D-76128 Karlsruhe, Germany

Revised  November 2001 Published  January 2002

The paper starts with a short review on the history of PDEs and their related inequalities. We present a unified approach to the basic parabolic differential inequalities. It starts in Section 2 with the famous Lemma of Nagumo and leads to recent and new result on parabolic equations with singular nonlinear elliptic operators (7) that include the p-Laplacian and the operator of capillary surfaces. Sections 3 and 4 deal with the one-dimensional case and radial solutions. A Semi-Strong Minimum Principle is found in Section 6.
Keywords: p-Laplacian., nonlinear inequalities, Minimum principles, Parabolic differential equations, radial solutions.
Mathematics Subject Classification: Primary: 35K10; Secondary: 35K2.
Citation: Wolfgang Walter. Nonlinear parabolic differential equations and inequalities. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 451-468. doi: 10.3934/dcds.2002.8.451
[1]

Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447
[2]

Trad Alotaibi, D. D. Hai, R. Shivaji. Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4655-4666. doi: 10.3934/cpaa.2020131
[3]

Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593
[4]

Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623
[5]

Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731
[6]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[7]

Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111
[8]

Junyong Eom, Ryuichi Sato. Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199
[9]

Olga Salieva. On nonexistence of solutions to some nonlinear parabolic inequalities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 843-853. doi: 10.3934/cpaa.2017040
[10]

C. Fabry, Raul Manásevich. Equations with a $p$-Laplacian and an asymmetric nonlinear term. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 545-557. doi: 10.3934/dcds.2001.7.545
[11]

Xiaohong Li, Fengquan Li. Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities. Communications on Pure & Applied Analysis, 2012, 11 (3) : 935-943. doi: 10.3934/cpaa.2012.11.935
[12]

Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021033
[13]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[14]

Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002
[15]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4805-4821. doi: 10.3934/dcds.2021058
[16]

Shanming Ji, Jingxue Yin, Yutian Li. Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2411-2439. doi: 10.3934/dcds.2018100
[17]

Petru Jebelean. Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267
[18]

Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87
[19]

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
[20]

José A. Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel. Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1027-1050. doi: 10.3934/dcdsb.2006.6.1027

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (449)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy