【摘要】 研究了黎曼流形上双曲型方程未知阻尼系数的识别问题,对于具有初边界值的双曲型方程■²_tu（x,t）-Δ_gu（x,t）+p（x）■_tu（x,t）=0,x∈M,0<t<T,其中M是边界为M的可微分流形,初始条件是u（x,0）=u₀,■_tu（x,0）=u₁,在边界Σ=■M×（0,T）上,u（x,t）=h,u₀,u₁,h都是给定的。令u=u_p1,0-u_p2,0,p₀=p₁,f=p₂-p₁,R=■_tu_p2,0,即可得到反问题■²_tu（x,t）-Δ_gu（x,t）+p₀（x）■_tu（x,t）=f（x） R（x,t）,x∈M,0<t<T,初始条件为u（x,0）=■_tu（x,0）=0,在边界Σ=■M×（0,T）上,u（x,t）=0。将该反问题延拓到（-T,T）,通过引入截断函数,给出了黎曼流形上的卡尔曼估计,对未知系数p₀（x）进行能量估计,探讨了反问题的Lipschitz稳定性。更多还原

【Abstract】 The problem of identifying unknown damping coefficient of hyperbolic equations on Riemannian manifolds is studied.For an initial boundary value problem of hyperbolic equation ■²_tu（x,t）-Δ_gu （x,t）+p（x）■_tu（x,t）=0,x∈M,0<t<T,where M is a differentiable manifold with boundary ■M,the initial conditions are u（x,0）=u₀,,■_tu（x,0）=u₁,the boundary condition is u（x,t）=h in Σ=■M×（0,T）,u₀,u₁,h are known.Set u=u_p1,0-u_p2,0,p₀=p₁,f=p₂-p₁,R=■_tu_p2,0,the inverse problem ■²_tu（x,t）-Δ_gu（x,t）+p₀（x）■_tu（x,t）=f（x） R（x,t）,x∈M,0<t<T can be obtained,the initial values are u（x,0）=■_tu（x,0）=0,the boundary condition is u（x,t）=0 in Σ=■M×（0,T）.First the inverse problem is extended to （-T,T）,then by introducing a cut off function,the Carleman estimate on the Riemannian manifold is given.The energy estimate of unknown coefficient p₀（x） is established,and the Lipschitz stability of inverse problem is discussed.更多还原