A Stabilization Method for the Tricomi Problem

Abstract

We prove the existence of a generalized solution of the Tricomi problem for the equation $L_0[u]=T[u]+\lambda l(u):=y{u}_{zz}+u_{yy}+\lambda 1(u)=f$, where $l={\alpha }^1\partial /\partial x+{\alpha }^2\partial /\partial y$ is a special differential operator and $\lambda \ge 0$ is a constant. Then we show the solvability of an initial boundary value problem for the evolution equation $L[u]=T[u]+\partial (u)/\partial t=F$ by an aproximation method. It is shown that the generalized solution of the evolution problem converges to the generalized solution of the Tricomi problem $T[u]=f$ as $t\to \infty$. The rate of convergence is estimated.