In this paper, we obtain the existence
result of viscosity solutions to the initial and boundary value problem for a
nonlinear degenerate parabolic inhomogeneous equation of the form , where denotes infinity Laplacian given by .
References
[1]
Aronsson, G. (1965) Minimization Problems for the Functional. Arkiv for Matematik, 6, 33-53. http://dx.doi.org/10.1007/BF02591326
[2]
Aronsson, G. (1966) Minimization Problems for the Functional. II. Arkiv for Matematik, 6, 409-431.
[3]
Crandall, M.G., Evans, L.C. and Gariepy, R.F. (2001) Optimal Lipschitz Extensions and the Infinity Laplacian. Calculus of Variations and Partial Differential Equations, 13, 123-139.
[4]
Jensen, R. (1993) Uniqueness of Lipschitz Extensions: Minimizing the Sup Norm of the Gradient. Archive for Rational Mechanics and Analysis, 123, 51-74. http://dx.doi.org/10.1007/BF00386368
[5]
Aronsson, G., Crandall, M. and Juutinen, P. (2004) A Tour of the Theory of Absolute Minimizing Functions. Bulletin of the AMS, 41, 439-505. http://dx.doi.org/10.1090/S0273-0979-04-01035-3
[6]
Juutinen, P. and Kawohl, B. (2006) On the Evolution Governed by the Infinity Laplacian. Mathematische Annalen, 335, 819-851. http://dx.doi.org/10.1007/s00208-006-0766-3
[7]
Liu, F. and Yang, X.P. (2015) Viscosity Solutions to a Parabolic Inhomogeneous Equation Associated with Infinity Laplacian. Acta Mathematica Sinica, English Series, 31, 255-271. http://dx.doi.org/10.1007/s10114-015-3244-6
[8]
Peres, Y., Schramm, O., Sheffield, S. and Wilson, D. (2009) Tug of War and the Infinity Laplacian. Journal of the American Mathematical Society, 22, 167-210. http://dx.doi.org/10.1090/S0894-0347-08-00606-1
[9]
Akagi, G. and Suzuki, K. (2007) On a Certain Degenerate Parabolic Equation Associated with the Infinity-Laplacian. Discrete and Continuous Dynamical Systems, Supplement, 18-27.
[10]
Akagi, G. and Suzuki, K. (2008) Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian. Calculus of Variations and Partial Differential Equations, 31, 457-471. http://dx.doi.org/10.1007/s00526-007-0117-6
[11]
Akagi, G., Juutinen, P. and Kajikiya, R. (2009) Asymptotic Behavior of Viscosity Solutions for a Degenerate Parabolic Equation Associated with the Infinity-Laplacian. Mathematische Annalen, 343, 921-953. http://dx.doi.org/10.1007/s00208-008-0297-1
[12]
Laurencot, P. and Stinner, C. (2010) Refined Asymptotics for the Infinite Heat Equation with Homogeneous Dirichlet Boundary Conditions. Communications in Partial Differential Equations, 36, 532-546. http://dx.doi.org/10.1080/03605302.2010.498493
[13]
Portilheiro, M. and Vázquez, J.L. (2012) Degenerate Homogeneous Parabolic Equations Associated with the Infinity-Laplacian. Calculus of Variations and Partial Differential Equations, 31, 457-471. http://dx.doi.org/10.1007/s00526-012-0500-9
[14]
Portilheiro, M. and Vázquez, J.L. (2012) A Porous Medium Equation Involving the Infinity-Laplacian, Viscosity Solutions and Asymptotic Behaviour. Communications in Partial Differential Equations, 37, 753-793. http://dx.doi.org/10.1080/03605302.2012.662665
[15]
Caselles, V., Morel, J.M. and Sbert, C. (1998) An Axiomatic Approach to Image Interpolation. IEEE Transactions on Image Processing, 7, 376-386. http://dx.doi.org/10.1109/83.661188
[16]
Crandall, M.G., Ishii, H. and Lions, P.L. (1992) User’s Guide to Viscosity Solutions of Second-Order Partial Differential Equations. Bulletin of the AMS, 27, 1-67. http://dx.doi.org/10.1090/S0273-0979-1992-00266-5
[17]
Ladyzenskaya, O.A., Solonnikov, V.A. and Ural’ceva, N.N. (1967) Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I.