Uniqueness of the solution of a partial differential equation problem with a non-constant coefficient

Abstract:

We consider the problem $K(x)u_{xx}=u_{t}$ , $0<x<1$, $t\geq 0$, where $K(x)$ is bounded below by a positive constant. The solution on the boundary $x=0$ is a known function and $u_{x}(0,t)=0$. This is an ill-posed problem in the sense that a small disturbance on the boundary specification can produce a big change in its solution, if it exists. In a previous work, we used a Wavelet Galerkin Method with the Meyer Multiresolution Analysis to generate a sequence of well-posed approximating problems to it. In the present work, by assuming that $1/K(x)$ is Lipschitz, we are able to prove that the existence of a solution $u(x,\cdot )\in H^{1}(R)$, for this problem, implies its uniqueness.