<p>This paper studies parabolic and elliptic equations of Arrhenius type arising in chemical kinetics. We first show that the fundamental Arrhenius equation posed in a bounded domain with smooth boundary, subject to homogeneous Dirichlet or Neumann boundary conditions and sufficiently smooth initial data, admits bounded classical solutions; in particular, solutions do not blow up in finite time. We then derive necessary and sufficient conditions, formulated in terms of the behavior of a modified Arrhenius reaction term, that ensure boundedness of solutions. A blow-up criterion for the Arrhenius-Kooij equation is also established. In the second part of the paper, we study the corresponding elliptic equations. To provide a comprehensive perspective, we recall their connection with the classical Liouville approach, rooted in the theory of surfaces with constant Gaussian curvature. As an application, we construct several explicit regular solutions to the associated boundary value problems using the Liouville formula, thereby recovering known results previously obtained by various authors via different methods. This approach also opens new possibilities for analyzing boundary value problems in a broad class of planar domains, not necessarily symmetric, through the use of analytic function techniques.</p>