-
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
Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation
College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China |
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.
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. |
[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. |
[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. |
[4] |
G. Aronsson,
Minimization problems for the functional supxF(x, f(x), f'(x)),, Ark. Mat., 6 (1965), 33-53.
doi: 10.1007/BF02591326. |
[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. |
[6] |
G. Aronsson,
Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561.
doi: 10.1007/BF02591928. |
[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. |
[8] |
F. Cao, Approximations dequations paraboliques par des schemas invariants; une theorie de linterpolation; applications au traitement dimages, PhD Thesis (in English), 2000. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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). |
[16] |
P. Lindqvist, Notes on the Infinity Laplace Equation(SpringerBriefs in Mathematics), Springer, (2016).
doi: 10.1007/978-3-319-31532-4. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
G. Aronsson,
Minimization problems for the functional supxF(x, f(x), f'(x)),, Ark. Mat., 6 (1965), 33-53.
doi: 10.1007/BF02591326. |
[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. |
[6] |
G. Aronsson,
Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561.
doi: 10.1007/BF02591928. |
[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. |
[8] |
F. Cao, Approximations dequations paraboliques par des schemas invariants; une theorie de linterpolation; applications au traitement dimages, PhD Thesis (in English), 2000. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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). |
[16] |
P. Lindqvist, Notes on the Infinity Laplace Equation(SpringerBriefs in Mathematics), Springer, (2016).
doi: 10.1007/978-3-319-31532-4. |
[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. |
[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. |
[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. |
[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 and Continuous Dynamical Systems, 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 and 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 and 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 and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 |
[6] |
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 |
[7] |
Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations and Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026 |
[8] |
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 |
[9] |
Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013 |
[10] |
Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 |
[11] |
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 and Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053 |
[12] |
Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017 |
[13] |
Jesus Ildefonso Díaz, Jacqueline Fleckinger-Pellé. Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 193-200. doi: 10.3934/dcds.2004.10.193 |
[14] |
Pierpaolo Soravia. Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation. Mathematical Control and Related Fields, 2012, 2 (4) : 399-427. doi: 10.3934/mcrf.2012.2.399 |
[15] |
Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure and Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69 |
[16] |
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 |
[17] |
Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 |
[18] |
Francisco Ortegón Gallego, María Teresa González Montesinos. Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system. Communications on Pure and Applied Analysis, 2007, 6 (1) : 23-42. doi: 10.3934/cpaa.2007.6.23 |
[19] |
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 and Applied Analysis, 2016, 15 (1) : 197-217. doi: 10.3934/cpaa.2016.15.197 |
[20] |
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 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]