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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

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