A NOTE ON GRADIENT BLOWUP RATE OF THE INHOMOGENEOUS HAMILTON-JACOBI EQUATIONS

doi:10.1016/S0252-9602(13)60029-6

数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (3): 678-686.doi: 10.1016/S0252-9602(13)60029-6

A NOTE ON GRADIENT BLOWUP RATE OF THE INHOMOGENEOUS HAMILTON-JACOBI EQUATIONS

张正策|李振杰

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

收稿日期:2011-06-16 修回日期:2011-09-11 出版日期:2013-05-20 发布日期:2013-05-20
基金资助:
The first author is supported by Youth Foundation of NSFC (10701061), Fundamental Research Funds for the Central Universities of China, and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

A NOTE ON GRADIENT BLOWUP RATE OF THE INHOMOGENEOUS HAMILTON-JACOBI EQUATIONS

ZHANG Zheng-Ce, LI Zhen-Jie

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received:2011-06-16 Revised:2011-09-11 Online:2013-05-20 Published:2013-05-20
Supported by:
The first author is supported by Youth Foundation of NSFC (10701061), Fundamental Research Funds for the Central Universities of China, and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

摘要/Abstract

摘要：

The gradient blowup of the equation u_t = Δu + a(x)|∇u|_p + h(x), where p > 2, is studied. It is shown that the gradient blowup rate will never match that of the self-similar variables. The exact blowup rate for radial solutions is established under the assumptions on the initial data so that the solution is monotonically increasing in time.

关键词: Gradient blowup, Hamilton-Jacobi equation, inhomogeneous

Abstract:

Key words: Gradient blowup, Hamilton-Jacobi equation, inhomogeneous

中图分类号:

35B35

张正策, 李振杰. A NOTE ON GRADIENT BLOWUP RATE OF THE INHOMOGENEOUS HAMILTON-JACOBI EQUATIONS[J]. 数学物理学报(英文版), 2013, 33(3): 678-686.

ZHANG Zheng-Ce, LI Zhen-Jie. A NOTE ON GRADIENT BLOWUP RATE OF THE INHOMOGENEOUS HAMILTON-JACOBI EQUATIONS[J]. Acta mathematica scientia,Series B, 2013, 33(3): 678-686.

参考文献

[1] Conner G R, Grant C P. Asymptotics of blow-up for a convection-diffusion equation with conservation. Differential Integral Equations, 1996, 9: 719–728

[2] Guo J S, Hu B. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete Contin Dyn Sys, 2008, 20: 927–937

[3] Kardar M, Parise G, Zhang Y C. Dynamic scaling of growing interfaces. Phys Rev Lett, 1986, 56: 889–892

[4] Krug J, Spohn H. Universality classes for deterministic surface growth. Phys Rev A, 1988, 38: 4271–4283

[5] Lieberman G M. Second Order Parabolic Differential Equations. Singapore: World Scientific, 1996

[6] Lions P L. Generalized solutions of Hamilton-Jacobi Equations. Pitman Research Notes inMath 69. Boston: Mass-London, 1982

[7] Quittner P, Souplet Ph. Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States. Basel: Birkh¨auser Verlag AG, 2007

[8] Souplet Ph, Zhang Q. Global solutions of inhomogeneous Hamilton-Jacobi equations. J D’Analyse Math, 2006, 99: 355–396

[9] Zhang Z C, Hu B. Gradient blowup rate for a semilinear parabolic equation. Discrete Contin Dyn, Sys, 2010, 26: 767–779

[10] Zhang Z C, Hu B. Rate estimate of gradient blowup for a heat equation with exponential nonlinearity. Nonlinear Analysis, 2010, 72: 4594–4601

[11] Zhang Z C, Li Y Y. Gradient blowup solutions of a semilinear parabolic equation with exponential source. Comm Pure Appl Anal, 2013, 12: 269–280

[12] Zhang Z C, Li Y Y. Boundedness of global solutions for a heat equation with exponential gradient source. Abstract Appl Anal, 2012, 2012: 1–10

[13] Zhang Z C, Wang B. Blow-up rate estimate parabolic equation with nonlinear gradient term. Appl Math Mech -Engl Ed, 2010, 31: 787–796

[14] Zhu L P, Zhang Z C. Rate of approach to the steady state for a diffusion-convection equation on annular domains. E J Qualitative Theory Diff Equations, 2012, 39: 1–10

A NOTE ON GRADIENT BLOWUP RATE OF THE INHOMOGENEOUS HAMILTON-JACOBI EQUATIONS

A NOTE ON GRADIENT BLOWUP RATE OF THE INHOMOGENEOUS HAMILTON-JACOBI EQUATIONS

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