Existence of weak solutions for a parabolic elliptic-hyperbolic Tricomi problem

Abstract

It is well-known that the pioneer of mixed type boundary value problems is F. G. Tricomi (1923) with his Tricomi equation: $yu_{xx}+u_{yy}=0$. In this paper we consider the more general case of above equation so that \[ Lu\equiv K_{1}(y)u_{xx}+(K_{2}(y)u_{y})^{\prime}+ru=f \] is hyperbolic-elliptic and parabolic, and then prove the existence of weak solutions for the corresponding Tricomi problem by employing the well-known a-b-c energy integral method to establish an a-priori estimate. This result is interesting in fluid mechanics.