  Our concerns here are blow-up solutions for ODEs with exponential
nonlinearity from the viewpoint of dynamical systems and their numerical
validations. As an example, the finite difference discretization of $u_t =
u_{xx} + e^{u^m}$ with the homogeneous Dirichlet boundary condition is
considered. Our idea is based on compactification of phase spaces and
time-scale desingularization as in previous works. In the present case,
treatment of exponential nonlinearity is the main issue. Fortunately, under a
kind of exponential homogeneity of vector field, we can treat the problem in
the same way as polynomial vector fields. In particular, we can characterize
and validate blow-up solutions with their blow-up times for differential
equations with such exponential nonlinearity in the similar way to previous
works. A series of technical treatments of exponential nonlinearity in blow-up
problems is also shown with concrete validation examples.
