Boundary and internal layers in a semilinear parabolic problem

For a singular perturbation problem of a certain class of reaction diffusion equations with space-dependent nonlinearities, we compared the effect that the presence of boundary layers has on the existence and stability of stationary solutions versus that of the internal layers.;We first extended the classical Sturm-Liouville theory to treat this problem. Then we computed the number of stable solutions for a type of nonlinearity in which the space dependence is given by a step function. We also completely determined the attractor for a few particular cases of this nonlinearity and we were able to compute the relevant curves numerically. Also for this case, we showed that our results are robust and that, in particular the structure of these attractors persist under small perturbations.;For a more general type of nonlinearities, we constructed an approximate solution that allowed us to determined the sign of the critical eigenvalues for the boundary layers. This result is made rigorous through the use of a Liapunov-Schmidt decomposition.