American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Fredholm theory for an elliptic differential operator defined on $ \mathbb{R}^n $ and acting on generalized Sobolev spaces
  • CPAA Home
  • This Issue
  • Next Article
    Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data
March  2020, 19(3): 1449-1462. doi: 10.3934/cpaa.2020071

Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation

Gang Li , Fen Gu and  Feida Jiang ,

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author

Received  April 2019 Revised  August 2019 Published  November 2019

Fund Project: This work was supported by National Natural Science Foundation of China (No. 11771214)

In this paper, we investigate positive viscosity solutions of a third degree homogeneous parabolic equation $ u^{2}u_{t} = \Delta_{\infty}u $. We prove a comparison principle, existence and uniqueness of continuous positive viscosity solutions.

Keywords: Parabolic infinity Laplacian equation, viscosity solution, comparison principle, existence.
Mathematics Subject Classification: Primary: 35K55, 35K65; Secondary: 35D40.
Citation: Gang Li, Fen Gu, Feida Jiang. Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1449-1462. doi: 10.3934/cpaa.2020071
References:
[1]

G. Akagi, P. Juutinen and R. Kajikiya, Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian, Math. Annalen, 343 (2009), 921-953.  doi: 10.1007/s00208-008-0297-1.  Google Scholar

[2]

G. Akagi and K. Suzuki, On a certain degenerate parabolic equation associated with the infinity-Laplacian, Discrete Contin. Dyn. Syst., (supplement), (2007), 18–27.  Google Scholar

[3]

G. Akagi and K. Suzuki, Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian, Calc. Var. and Partical Differential Equations, 31 (2008), 457-471.  doi: 10.1007/s00526-007-0117-6.  Google Scholar

[4]

G. Aronsson, Minimization problems for the functional supxF(x, f(x), f'(x)),, Ark. Mat., 6 (1965), 33-53.  doi: 10.1007/BF02591326.  Google Scholar

[5]

G. Aronsson, Minimization problems for the functional supxF(x, f(x), f'(x)), Ⅱ, Ark. Mat., 6 (1966), 409-431.  doi: 10.1007/BF02590964.  Google Scholar

[6]

G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561.  doi: 10.1007/BF02591928.  Google Scholar

[7]

G. Aronsson, M. Crandall and P. Juutien, A tour of the theory of absolute minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505.  doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[8]

F. Cao, Approximations dequations paraboliques par des schemas invariants; une theorie de linterpolation; applications au traitement dimages, PhD Thesis (in English), 2000. Google Scholar

[9]

V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386.  doi: 10.1109/83.661188.  Google Scholar

[10]

M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. and Partical Differential Equations, 13 (2001), 123-139.   Google Scholar

[11]

M. G. Crandall, H. Ishii and P.-L Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[12]

M. G. Crandall and P. Wang, Another way to say caloric. Dedicated to Philippe Benilan, J. Evol. Equ., 3 (2003), 653-672.  doi: 10.1007/s00028-003-0146-3.  Google Scholar

[13]

R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 1-74.  doi: 10.1007/BF00386368.  Google Scholar

[14]

P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851.  doi: 10.1007/s00208-006-0766-3.  Google Scholar

[15]

O. A. Ladyženskaya, V. A. Solommikov and N. N. Urall'ceva, Linear and Quasilinear Equations of Parabolic Type, in Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R. I., (1967).  Google Scholar

[16]

P. Lindqvist, Notes on the Infinity Laplace Equation(SpringerBriefs in Mathematics), Springer, (2016). doi: 10.1007/978-3-319-31532-4.  Google Scholar

[17]

M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the $p$-Laplace diffusion equation, Comm. Partial Differential Equations, 22 (1997), 381-411.  doi: 10.1080/03605309708821268.  Google Scholar

[18]

M. Portilheiro and J. L. Vázquez, A porous medium equation involving the infinity-Laplacian, Viscosity solutions and asymptotic behaviour, Comm. Partial Differential Equations, 37 (2012), 753-793.  doi: 10.1080/03605302.2012.662665.  Google Scholar

[19]

M. Portilheiro and J. L. Vázquez, Degenerate homogeneous parabolic equations associated with the infinity-Laplacian, Calc. Var. and Partial Differential Equations, 31 (2012), 457-471.  doi: 10.1007/s00526-012-0500-9.  Google Scholar

show all references

References:
[1]

G. Akagi, P. Juutinen and R. Kajikiya, Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian, Math. Annalen, 343 (2009), 921-953.  doi: 10.1007/s00208-008-0297-1.  Google Scholar

[2]

G. Akagi and K. Suzuki, On a certain degenerate parabolic equation associated with the infinity-Laplacian, Discrete Contin. Dyn. Syst., (supplement), (2007), 18–27.  Google Scholar

[3]

G. Akagi and K. Suzuki, Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian, Calc. Var. and Partical Differential Equations, 31 (2008), 457-471.  doi: 10.1007/s00526-007-0117-6.  Google Scholar

[4]

G. Aronsson, Minimization problems for the functional supxF(x, f(x), f'(x)),, Ark. Mat., 6 (1965), 33-53.  doi: 10.1007/BF02591326.  Google Scholar

[5]

G. Aronsson, Minimization problems for the functional supxF(x, f(x), f'(x)), Ⅱ, Ark. Mat., 6 (1966), 409-431.  doi: 10.1007/BF02590964.  Google Scholar

[6]

G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561.  doi: 10.1007/BF02591928.  Google Scholar

[7]

G. Aronsson, M. Crandall and P. Juutien, A tour of the theory of absolute minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505.  doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[8]

F. Cao, Approximations dequations paraboliques par des schemas invariants; une theorie de linterpolation; applications au traitement dimages, PhD Thesis (in English), 2000. Google Scholar

[9]

V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386.  doi: 10.1109/83.661188.  Google Scholar

[10]

M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. and Partical Differential Equations, 13 (2001), 123-139.   Google Scholar

[11]

M. G. Crandall, H. Ishii and P.-L Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[12]

M. G. Crandall and P. Wang, Another way to say caloric. Dedicated to Philippe Benilan, J. Evol. Equ., 3 (2003), 653-672.  doi: 10.1007/s00028-003-0146-3.  Google Scholar

[13]

R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 1-74.  doi: 10.1007/BF00386368.  Google Scholar

[14]

P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851.  doi: 10.1007/s00208-006-0766-3.  Google Scholar

[15]

O. A. Ladyženskaya, V. A. Solommikov and N. N. Urall'ceva, Linear and Quasilinear Equations of Parabolic Type, in Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R. I., (1967).  Google Scholar

[16]

P. Lindqvist, Notes on the Infinity Laplace Equation(SpringerBriefs in Mathematics), Springer, (2016). doi: 10.1007/978-3-319-31532-4.  Google Scholar

[17]

M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the $p$-Laplace diffusion equation, Comm. Partial Differential Equations, 22 (1997), 381-411.  doi: 10.1080/03605309708821268.  Google Scholar

[18]

M. Portilheiro and J. L. Vázquez, A porous medium equation involving the infinity-Laplacian, Viscosity solutions and asymptotic behaviour, Comm. Partial Differential Equations, 37 (2012), 753-793.  doi: 10.1080/03605302.2012.662665.  Google Scholar

[19]

M. Portilheiro and J. L. Vázquez, Degenerate homogeneous parabolic equations associated with the infinity-Laplacian, Calc. Var. and Partial Differential Equations, 31 (2012), 457-471.  doi: 10.1007/s00526-012-0500-9.  Google Scholar

[1]

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
[2]

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 & Continuous Dynamical Systems - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683
[3]

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

Jacques Giacomoni, Tuhina Mukherjee, Konijeti Sreenadh. Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 311-337. doi: 10.3934/dcdss.2019022
[5]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[6]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
[7]

Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114
[8]

Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013
[9]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
[10]

Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053
[11]

Jesus Ildefonso Díaz, Jacqueline Fleckinger-Pellé. Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 193-200. doi: 10.3934/dcds.2004.10.193
[12]

Pierpaolo Soravia. Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation. Mathematical Control & Related Fields, 2012, 2 (4) : 399-427. doi: 10.3934/mcrf.2012.2.399
[13]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[14]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69
[15]

Francisco Ortegón Gallego, María Teresa González Montesinos. Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system. Communications on Pure & Applied Analysis, 2007, 6 (1) : 23-42. doi: 10.3934/cpaa.2007.6.23
[16]

Dominique Blanchard, Olivier Guibé, Hicham Redwane. Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 197-217. doi: 10.3934/cpaa.2016.15.197
[17]

Pelin G. Geredeli, Azer Khanmamedov. Long-time dynamics of the parabolic $p$-Laplacian equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 735-754. doi: 10.3934/cpaa.2013.12.735
[18]

Baojun Bian, Shuntai Hu, Quan Yuan, Harry Zheng. Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5413-5433. doi: 10.3934/dcds.2015.35.5413
[19]

Mehdi Badra, Kaushik Bal, Jacques Giacomoni. Existence results to a quasilinear and singular parabolic equation. Conference Publications, 2011, 2011 (Special) : 117-125. doi: 10.3934/proc.2011.2011.117
[20]

Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (18)
  • HTML views (13)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences