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The initial and Neumann boundary value problem for a class parabolic Monge-Ampère equation. (English) Zbl 1293.35159

Summary: We consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class nonlinear parabolic equation of Monge-Ampère type. We show that such solution exists for all times and is unique. It converges eventually to a solution that satisfies a Neumann type problem for nonlinear elliptic equation of Monge-Ampère type.

MSC:

35K96 Parabolic Monge-Ampère equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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[1] Cossette, J.-F.; Smolarkiewicz, P. K., A Monge-Ampère enhancement for semi-Lagrangian methods, Computers & Fluids, 46, 180-185 (2011) · Zbl 1433.76130 · doi:10.1016/j.compfluid.2011.01.029
[2] Liu, Z.; He, Y., Solving the elliptic Monge-Ampère equation by Kansa’s method, Engineering Analysis with Boundary Elements, 37, 1, 84-88 (2013) · Zbl 1352.65570 · doi:10.1016/j.enganabound.2012.09.004
[3] Hong, J. X., The global smooth solutions of Cauchy problems for hyperbolic equation of Monge-Ampère type, Nonlinear Analysis: Theory, Methods & Applications, 24, 12, 1649-1663 (1995) · Zbl 0830.35082 · doi:10.1016/0362-546X(94)00248-G
[4] Dean, E. J.; Glowinski, R., Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type, Computer Methods in Applied Mechanics and Engineering, 195, 13-16, 1344-1386 (2006) · Zbl 1119.65116 · doi:10.1016/j.cma.2005.05.023
[5] Gutiérrez, C. E., The Monge-Ampère Equation (2011), Basel, Switzerland: Birkhäuser, Basel, Switzerland
[6] Schnürer, O. C.; Smoczyk, K., Neumann and second boundary value problems for Hessian and Gauss curvature flows, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 20, 6, 1043-1073 (2003) · Zbl 1032.53058 · doi:10.1016/S0294-1449(03)00021-0
[7] Lions, P.-L.; Trudinger, N. S.; Urbas, J. I. E., The Neumann problem for equations of Monge-Ampère type, Communications on Pure and Applied Mathematics, 39, 4, 539-563 (1986) · Zbl 0604.35027 · doi:10.1002/cpa.3160390405
[8] Urbas, J., The second boundary value problem for a class of Hessian equations, Communications in Partial Differential Equations, 26, 5-6, 859-882 (2001) · Zbl 1194.35158 · doi:10.1081/PDE-100002381
[9] Urbas, J., Oblique boundary value problems for equations of Monge-Ampère type, Calculus of Variations and Partial Differential Equations, 7, 1, 19-39 (1998) · Zbl 0912.35068 · doi:10.1007/s005260050097
[10] Urbas, J., On the second boundary value problem for equations of Monge-Ampère type, Journal für die Reine und Angewandte Mathematik, 487, 115-124 (1997) · Zbl 0880.35031 · doi:10.1515/crll.1997.487.115
[11] Zhou, W. S.; Lian, S. Z., The third initial-boundary value problem for an equation of parabolic Monge-Ampère type, Journal of Jilin University, 1, 23-30 (2001) · Zbl 0993.35040
[12] Kuang, J. C., Applied Inequalities (2004), Shandong Science and Technology Press
[13] Evans, L. C., Partial Differential Equations. Partial Differential Equations, Graduate Studies in Mathematics, 19, xviii+662 (1998), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0902.35002
[14] Chen, Y. Z., Some methods of Krylov for the a priori estimation of solutions of fully nonlinear equations, Advances in Mathematics, 15, 1, 63-101 (1986) · Zbl 0622.35018
[15] Dong, G. C., Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations, Journal of Partial Differential Equations. Series A, 1, 2, 12-42 (1988) · Zbl 0699.35152
[16] Ladyženskaja, O. A.; Solonnikov, V. A.; Ural’zeva, N. N., Linear and Quasilinear Equations of Parabolic Type (1995), American Mathematical Society
[17] Gerhardt, C., Hypersurfaces of prescribed curvature in Lorentzian manifolds, Indiana University Mathematics Journal, 49, 3, 1125-1153 (2000) · Zbl 1034.53064 · doi:10.1512/iumj.2000.49.1861
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