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

Authors: Ernesto Prado Lopes and José Roberto Linhares de Mattos
Journal: Proc. Amer. Math. Soc. 136 (2008), 2201-2207
MSC (2000): Primary 65T60
Published electronically: February 12, 2008
MathSciNet review: 2383526
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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.

References [Enhancements On Off] (What's this?)

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Additional Information

Ernesto Prado Lopes
Affiliation: Institute of Mathematics, Tecnology Center, Bloco C and COPPE, Systems and Computing Engineering Program, Tecnology Center, Bloco H, Federal University of Rio de Janeiro, Ilha do Fundão, Rio de Janeiro, RJ, CEP 21945-970, Brazil

José Roberto Linhares de Mattos
Affiliation: Department of Geometry, Institute of Mathematics, Fluminense Federal University, Rua Mário Santos Braga s/n, Valonguinho, Niterói, Rio de Janeiro, RJ, CEP 24020-140, Brazil

Received by editor(s): October 18, 2006
Published electronically: February 12, 2008
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2008 American Mathematical Society