三维空间中非线性波方程的整体与局部C^2(英文)

GLOBAL AND LOCAL C 2 SOLUTION FOR NONLINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS

讨论下面方程的Ｃａｕｃｈｙ问题：ｕｔｔ－Δｕ＝｜ｕｔ（ｘ，ｔ）｜ｐ，ｔ≥０，ｘ∈Ｒ３，ｕ（ｘ，０）＝εｆ（ｘ），ｕｔ（ｘ，０）＝εｇ（ｘ），ｘ∈Ｒ３，这里Δ＝∑３ｉ＝１２ｘ２ｉ，常数ｐ＞１，ε是正参数，Ｈ．Ｔａｋａｍｕｒａ（ＣｏｍｍｉｎＰＤＥ，１９９２，１７（１＆２）：１８９）猜侧上面的Ｃａｕｃｈｙ问题在ｐ＞２时是否对充分小的初值存在整体Ｃ２解．本文将在ｆ（ｘ）。

This paper deals with the following Cauchy problemu tt -Δu=|u t(x,t)| p,t≥0,x∈R 3, u(x,0)=εf(x),u t(x,0)=εg(x),x∈R 3,where, Δ=∑3i=1 2 x 2 i, constant p>1 and ε is a positive parameter. H. Takamura(Comm in PDE, 1992,17(1&2):189) conjectured whether the above Cauchy problem in the case p>2 admits a global C 2 solution for small data. We shall partially answer this problem in the case p>3 with the Cauchy data f(x) and g(x) satisfying suitable assumptions.