Sufficient conditions are established for the exponential stability/instability of the equilibrium profiles for a linearized model of nonisothermal axial dispersion tubular reactors. The considered reactors are assumed to involve a chemical reaction of the form A -> B, where A and B denote the reactant and the product, respectively, and where the Peclet numbers appearing in the energy and mass balance PDEs are assumed to be equal. First, different kinds of linearization of infinite-dimensional dynamical systems are presented. Then the considered linearized model around any equilibrium is shown to be well-posed. Moreover, by using a Lyapunov-based approach, exponential stability is adressed. In the case when the reactor can exhibit only one equilibrium, it is shown that the latter is always exponentially stable. When three equilibrium profiles are exhibited, bistability is established, i.e. the stability pattern "(exponentially) stable -- unstable -- stable" is proven for the linearized model. The results are illustrated by some numerical simulations.