JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2018, Vol. 53 ›› Issue (8): 77-83.doi: 10.6040/j.issn.1671-9352.0.2017.341

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Comparison principles for viscosity solution of fully nonlinear parabolic equations with superlinear gradient nonlinearities

WANG Jun-fang, ZHAO Pei-hao   

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China
  • Received:2017-06-30 Online:2018-08-20 Published:2018-07-11

Abstract: A problem of fully nonlinear degenerate parabolic partial differential equations with a superlinear gradient nonlinearity is studied. A comparison result is proved between semicontinuous viscosity subsolutions and supersolutions having superlinear growth. We extend our result to monotone systems of parabolic equations.

Key words: degenerate parabolic equations, viscosity solution, comparison principle, monotone system, superlinear growth

CLC Number: 

  • O177
[1] ALVAREZ O. Bounded-from-below viscosity solutions of Hamilton-Jacobi equations[J]. Differential Integral Equations, 1997,10(3):419-436.
[2] ISHII H. Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above[J]. Applicable Analysis, 1997, 67(3-4):357-372.
[3] ISHII H. Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations[J]. Indiana University Mathematics Journal, 1984, 33(5):721-748.
[4] ISHII K, TOMITA Y. Unbounded viscosity solutions of nonlinear second order PDEs[J]. Advances in Mathematical Sciences and Applications, 2000, 10(2):689-710.
[5] AIZAWA S, TOMITA Y. Unbounded viscosity solutions of fully nonlinear elliptic equations in Rn[J]. Advances in Mathematical Sciences and Applications, 1993,2(2):297-316.
[6] KOIKE S, LEY O. Comparison principle for unbounded viscosity solutions of degenerate elliptic PDEs with gradient superlinear term[J]. Journal of Mathematical Analysis and Applications, 2001, 381(1):110-120.
[7] DA-LIO F, LEY O. Uniqueness results for second-order Bellman-Isaacs equations under quadratic growth assumptions and applications[J]. SIAM Journal on Control and Optimization, 2006, 45(1):74-106.
[8] DA-LIO F, LEY O. Convex Hamilton-Jacobi equations under superlinear growth conditions on data[J]. Applied Mathematics and Optimization, 2001, 63(3):309-339.
[9] BARLES G. Uniqueness and regularity results for first-order Hamilton-Jacobi equations[J]. Indiana University Mathematics Journal, 1990, 39(2):443-466.
[10] CRANDALL M G, ISHII H, LIONS P-L. Users guide to viscosity solutions of second order partial differential equations[J]. Bulletin of the American Mathematical Society, 1992, 27(1):1-67.
[11] CRANDALL M G, ISHII H, LIONS P-L. Viscosity solutions of Hamilton-Jacobi equations[J]. Transactions of the American Mathematical Society, 1983, 277(1):1-42.
[12] ISHII H, KOIKE S. Viscosity solutions for monotone systems of second-order elliptic PDEs[J]. Communications in Partial Differential Equations, 1991, 16(6):1095-1128.
[1] DAI Li-mei. Regularity of viscosity solutions to Hessian equations [J]. J4, 2010, 45(9): 62-64.
[2] WEI Li-Feng, CHEN Li. [J]. J4, 2009, 44(12): 1-5.
[3] HUANG Zong-Yuan, ZHANG Feng. Viscosity solutions of multidimensional quasilinear parabolic PDEs [J]. J4, 2008, 43(12): 5-9.
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