一类半线性波动方程解的渐近理论

The Asymptotic Theory for a Class of Wave Equations in Three Space Dimensions

研究了三维空间中如下半线性波动方程的初值问题 utt =Δu - b24 u+εF(u ,ε) , t >0 ,x∈R3,u(0 ,x ,ε) =u0 (x ,ε) ,ut(0 ,x ,ε) =u1(x ,ε) ,x∈R3,解的渐近理论。其中Δ = 3i=1 2 x2 i,常数b≥ 0。在古典空间C2 中得到了形式近似解的合理性在长时间t∈〔0 ,｜ε｜- 12 -k( p- 1) 〕(ε充分小 ,0 <2 -k(p- 1) <1,0 <k<1,p >3)

Deals with the asymptotic theory of initial value problems of semilinear wave equations in three space dimensions. u tt = Δ u-b 24u+εF(u,ε),t>0,x∈R 3, u(0,x,ε)=u 0(x,ε),u t(0,x,ε)=u 1(x,ε), x∈R 3. Where Δ =3i=1 2x 2 i, constant b>0. The validity of formal approximations on a long time scale of order |ε| -12-k(p-1) (ε is sufficiently small, 0<2-k(p-1)<1, 0<k<1,p>3) is discussed in the classical sense of C 2.