The authors are concerned with weak solutions of the following \(problem:\) \(\sigma\) \(u_ t=\Delta u+f(x,t)\) on \(\Omega^-\), \(\beta u_{tt}=\Delta u+g(x,t)\) on \(\Omega^+\), \(u^-=u^+\), \(u_ n^-=u_ n^+\) on \(\partial \Omega^-\), \(u(x,0)=U_ 0(x)\), \(x\in R^ 2\); \(u_ t(x,0)=U_ 1(x)\), \(x\in \Omega^+,\) where \(\sigma\), \(\beta\) are positive constants, \(\Omega^-\) a bounded domain in \(R^ 2\) with smooth boundary, \(\Omega^+=int(R^ 2\setminus \Omega^-)\), \(u^+\) and \(u^-\) the limits on \(\partial \Omega^-\) from the exterior and interior of \(\Omega^-\), respectively, and \(u_ n\) is the outward normal derivative with respect to \(\partial \Omega^- \). They study the existence and regularity of solutions. The existence proof is based on the method of singular perturbation in \(\Omega^-\) and on the Galerkin's method.