一类拟线性广义Euler—poisson方程的奇异Cauchy问题

The Singular Cauchy Problem of a Kind of General ized quasi—1 inear Euler—Poiss on Equat ion

设a,b为两个实数,a<b;记Ω={(x,y)/a≤y<x≤b}我们考察如下的拟线性广义Euler-poisson方程的奇异Cauchy问题:

In this paper, We discuss the singular Cauchy problem of a generalized quasi—linear Euler-possion equation as follows:where Ω={(x,y)|a≤y<x≤b}, a,b and in are constants and m>2,F(x, y,u)=som from i=1 to 2 f_i(x,y,u) and f_i(x,y,u) (i=1.2) is a continuous function on Ω×R.We obtain the following results:Theoreml Let f_i(x,y,u) satisfying:1) f_i(x,y,u) satisfies Lipschitz Condition with respect u2) there exist a continuous function A(x,y) and there are two positive numbers C and α(0≤α≤1) such that|f_2(x^(?),y^(?),u)|≤C(A(x,y)+|u|~α) then the singular Cauchy problem of generalized quasi—linear Euler—poisson equation (Ⅰ) has, at least, a regular solution.Theorem 2. Let F(x,y,u) be a continuous function on (?)×R. if there are two positive numbers c and α (0≤α≤1) and there is a Continuous function A(x,y) such that|F(x^(?),y^(?),u)|≤C(A(x^(?),y) +|u|~α) then the singular Cauchy problem (Ⅰ) has at least a regular solution.Theorem 3. Let F(x,y,u) be a continuous function on (?)×R if F(x,u) satisfies Lipschitz condition with respect u, then the singular Cauchy problem (Ⅰ) has a unique regular solution and for any u_0∈C((?)), the iterative sequenceu_n(x_0, y_0)=- integral from n=y_0 to x_0 integral from n=y to x_0 ν(x^(?),y^(?),x_0^(?),y_0)F(x^(?),y^(?)u_(n-1))dxdy converges uniformly to the solution on .