关于二阶线性双曲型方程的奇Cauchy问题

On the Singular Cauchy Problem for Linear Hyperbolic Equation of the Second Order

本文研究了如下的奇Cauchy问题:我们所得到的主要结果是:若y≠0时,a,b,c,f∈c^1,而且存在充分小的正数δ,成立估计式则当τ(x)≡0,v(x)≡0时,问题(1)(2)存在着唯一的正则解u(x,y)∈D_1[u]≡{u(x,y)|u=0(1)y^(3-m/2)}.若把关于f的条件改为D_2[u]≡{u(x,y)|u=O(1)y^(2-m/2)}.这时系数a,b,c在y→0^+时还允许有奇性,因此在0<m<1的情况下,我们推广了[3]的结果,若a,b,c,f∈C^1;τ。v∈C^3.而在0<m<1/2时,则问题(1)(2)存在着唯一的正则解.对于更一般的方程: y^mk(x,y)u_(yy)-u_(xx)+a(x,y)u_x+b(x,y)u_y+c(x,y)u+ +f(x,y)=0.y>0,0<m<2,k(x,y)>0也可以类似地得到上面的结果.

In this paper, we consider linear hyperbolic equation of the second order in two variables with parabolic degenerating line:(1)The initial data given on the degenerating line y=0 are(2)We obtain the following results:Theorem 1. If a, b, c,f e C1 for y≠0; τ(x)≡0,v(x)≡0 and there existsδ>0, such thatThen the regular solution u(x, y) ∈ D1 [u] ≡{u(x, y)|u=O(1)y3-m/2} of the Cauchy problem (1)(2) exists and is unique.If in theorem 1 the condition assumed for f in 0<y≤δ is replaced by(n=0, 1) and 0<m<1, then this u(x,y) δ D2[u]≡{u(x,y)|u=O(1)y2-m/2}. Here the coefficients a,b,c of equation (1) may be singular at y=0 and so, it generalizes the results in [3].Theorem 2. If a, b,c,f ∈ee C1; τ,v ∈ C3 and 0 < m<1/2, then the regular solution of the Cauchy prohlem (1)(2) exists and is unique.