一类具有奇异系数的弱双曲型方程的解的可微性及渐近性

ON THE DIFFERENTIABILITY THEOREM AND THE ASYMPTOTIC BEHAVIOUR OF SOLUTION OF A CLASS OF WEAKLY HYPERBOLIC EQUATIONS WITH SINGULAR COEFFICIENTS

本文讨论如下形式的方程((?)/(?)~t-it~ρD_x)(?)/(?)~t+it~ρD_x+(α+β)/t~α)u+α/t~α-(?)/(?)~t+α(α+β)/t^(2α)u=f(t,x) (1)x∈R^n,0<t≤T.ρ>0,α≥1的常数。α及β也是常数。方程在 t=O 有重特征。而低阶项的系数正好在 t=0 有奇异性。我们在方程的低阶项符合一定条件,且方程的特征根的重数与低阶项的奇异性的阶数满足一定关系时,给出了方程(1)的解的唯一性与可微性定理。并讨论了当 t→+0 时,解的渐近性态。

In this paper,we study the following equation((?)-it~ρD_x)((?)+i+ρD_x+(a+β)/t^a)u+a/t^a (?)+1/t^(2a)a(a+β)u=f(1)where ρ<0,α≥1,a and β are Constants,If ρ-a+1>and a>o,a_o is apositive constant which dependent β.we establish the differentiabily Theoremof solution of (1).and Study the asymptotic behaviour as t→+O of solutionof(1).