Ma. Herrero et al., GLOBAL EXISTENCE FOR REACTION-DIFFUSION SYSTEMS MODELING IGNITION, Archive for Rational Mechanics and Analysis, 142(3), 1998, pp. 219-251
The pair of parabolic equations u(t) = a Delta u + f(u,v), (1) v(t) =
b Delta b - f(u, v), (2) with a > 0 and b > 0 models the temperature a
nd concentration for an exothermic chemical reaction for which just on
e species controls the reaction rate f. Of particular interest is the
case where f(u, v)= ve(u), (3) which appears in the Frank-Kamenetskii
approximation to Arrhenius-type reactions, We show here that for a lar
ge choice of the nonlinearity f(u,v) in (1), (2) (including the model
case (3)), the corresponding initial-value problem for(1), (2) in the
whole space with bounded initial data has a solution which exists for
all times. Finite-time blow-up might occur, though, for other choices
of function f(ld, v), and we discuss here a linear example which stron
gly hints at such behaviour.