2007, 2007(Special): 18-27. doi: 10.3934/proc.2007.2007.18

On a certain degenerate parabolic equation associated with the infinity-laplacian

1. 

Department of Machinery and Control Systems, College of Systems Engineering and Science,, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama 337-8570, Japan

2. 

Daiwa Institute of Research, 15-6 Fuyuki, Koto-ku, Tokyo 135-8460, Japan

Received  September 2006 Revised  January 2007 Published  September 2007

The comparison, uniqueness and existence of viscosity solutions to the Cauchy-Dirichlet problem are proved for a degenerate parabolic equation of the form $u_t$ = $\Delta_(\infty)u$, where $\Delta_(\infty)$ denotes the so-called infinity-Laplacian given by $\Delta_(\infty)u$ = $\Sigma^(N)_(i,j=1) u_x_i u_x_j u_(x_i)_x_j$ . Our proof relies on a coercive regularization of the equation, barrier function arguments and the stability of viscosity solutions.
Citation: Goro Akagi, Kazumasa Suzuki. On a certain degenerate parabolic equation associated with the infinity-laplacian. Conference Publications, 2007, 2007 (Special) : 18-27. doi: 10.3934/proc.2007.2007.18
[1]

Gang Li, Fen Gu, Feida Jiang. Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1449-1462. doi: 10.3934/cpaa.2020071

[2]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225

[3]

Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114

[4]

Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060

[5]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure and Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617

[6]

Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683

[7]

Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022

[8]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

[9]

Alexandre Montaru. Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 231-256. doi: 10.3934/dcdsb.2014.19.231

[10]

M. Sango. Weak solutions for a doubly degenerate quasilinear parabolic equation with random forcing. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 885-905. doi: 10.3934/dcdsb.2007.7.885

[11]

Michael Winkler. Nontrivial ordered ω-limit sets in a linear degenerate parabolic equation. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 739-750. doi: 10.3934/dcds.2007.17.739

[12]

U. Biccari, V. Hernández-Santamaría, J. Vancostenoble. Existence and cost of boundary controls for a degenerate/singular parabolic equation. Mathematical Control and Related Fields, 2022, 12 (2) : 495-530. doi: 10.3934/mcrf.2021032

[13]

Jacques Giacomoni, Tuhina Mukherjee, Konijeti Sreenadh. Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 311-337. doi: 10.3934/dcdss.2019022

[14]

Pelin G. Geredeli, Azer Khanmamedov. Long-time dynamics of the parabolic $p$-Laplacian equation. Communications on Pure and Applied Analysis, 2013, 12 (2) : 735-754. doi: 10.3934/cpaa.2013.12.735

[15]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations and Control Theory, 2022, 11 (2) : 399-414. doi: 10.3934/eect.2021005

[16]

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

[17]

Baojun Bian, Shuntai Hu, Quan Yuan, Harry Zheng. Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5413-5433. doi: 10.3934/dcds.2015.35.5413

[18]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313

[19]

Shuyu Gong, Ziwei Zhou, Jiguang Bao. Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4921-4936. doi: 10.3934/cpaa.2020218

[20]

Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure and Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043

 Impact Factor: 

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]