The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain

Correspondence should be addressed to Gerardo Rubio, [email protected] Received 20 December 2010; Revised 18 March 2011; Accepted 19 April 2011 Academic Editor: Lukasz Stettner Copyright q 2011 Gerardo Rubio. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the Cauchy-Dirichlet problem in 0,∞ × D for a class of linear parabolic partial differential equations. We assume that D ⊂ R is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.