Acta mathematica scientia,Series B ›› 2003, Vol. 23 ›› Issue (1): 16-22.

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REMARKS ON THE LIFESPAN FOR THE SOLUTION TO NONLINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS

 YANG Han, LIU Fa-Gui   

  1. Department of Applied Mathematics, South West Jiaotong University, Chengdu 610031, China North China Institute of Water Conservancy and Hydroeletric Power, Zhengzhou 450045, China
  • Online:2003-01-06 Published:2003-01-06
  • Supported by:

    This work was supported by South-West Jiaotong University Foundation

Abstract:

The authors consider the Cauchy problem for the following nonlinear wave equations
?????
utt − u = |∇v|2,
vtt − v = (u)2,
u(0, x) = "u0(x), ut(0, x) = "u1(x),
v(0, x) = "v0(x), vt(0, x) = "v1(x),
where x ∈ R3, t ≥ 0, " > 0 is a small parameter, and obtain the sharp bounds for the lifespan of solution to (0.1). Specially, it is proved that there exist two constants C1 and C2, which are independent of ", then the lifespan T(") satisfies the folowing inequalities C1 ≤ lim"!0 " ln T(") ≤ C2.

Key words: Nonlinear wave equations, lifespan, spherical mean

CLC Number: 

  • 35L70
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