Singular perturbation problems and the Hamilton-Jacobi equation (Q1080676)

The asymptotic behaviour of the solution \(u_{\epsilon}(x)\) of the Dirichlet problem of the linear equation \[ \epsilon \Delta u+\sum^{m}_{i=1}b_ i(x)u_{x_ i}+c(x)u=0\quad in\quad \Omega,\quad u=\phi \quad on\quad \partial \Omega, \] is studied. Under assumptions given below this solution \(u_{\epsilon}(x)\) decreases exponentially inside \(\Omega\) as \(\epsilon\) \(\to 0\). These assumptions include that \(\phi (x)>0\), the coefficients \(b_ i(x)\) and c(x) are sufficiently smooth, the point \(x=0\) is of attracting type as t increase to \(\infty\), \(m=2\) and \(c(0)<0\). This solution converges exponentially to 0 for any x and decreases exponentially along characteristic curves.