一类半线性Keldysh型方程解的注记

A Note on the Solution of a Class of Keldysh-Type Equations

Keldysh方程是在研究跨音速管道流问题时导出的一个简化的数学模型,也是研究混合型偏微分方程模型的一个典型代表.对于其含有非零源项的退化双曲部分的初值问题,本文利用部分Fourier变换与ODE求解的办法给出了相应线性方程解的一个显式表达式及其全局一致估计,并在这个估计的基础上利用不动点定理建立了一类半线性问题的解的全局存在性.同时给出了解的奇性传播可以仅沿一支特征线传播的一个例子.

Keldysh equation is a mathematical model which was derived in studying fluid dynamics, and also is a typical example of mixed type partial differential equations. For the initial value problem of its degenerate hyperbolic part with nonzero source term,we find a solution by use of partial Fourier transformation and 0DE＇s method, and then derive its a global uniform estimate. Based on it and the fixed point theorem,we establish the global existence for solution of a class of semilinear problem. Meanwhile,we find an example to desmontrate the propogation of singularity which is only along one of the characteristics.