Global existence for reaction-diffusion systems modelling ignition

• Published in: Archive for rational mechanics and analysis 1998-07, Vol.142 (3), p.219-251
• Main Authors: HERRERO, M. A, LACEY, A. A, VELAZQUEZ, J. J. L
• Format: Article
• Language: English
• Subjects: Applied sciences; Combustion of gaseous fuels; Combustion. Flame; Energy; Energy. Thermal use of fuels; Exact sciences and technology; Mathematical methods in physics; Numerical approximation and analysis; Ordinary and partial differential equations, boundary value problems; Physics; Theoretical studies. Data and constants. Metering
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s002050050091

The pair of parabolic equations (ProQuest: Formulae and/or non-USASCII text omitted; see image) , (ProQuest: Formulae and/or non-USASCII text omitted; see image) , with a>0 and b>0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f. Of particular interest is the case where (ProQuest: Formulae and/or non-USASCII text omitted; see image) , which appears in the Frank-Kamenetskii approximation to Arrhenius-type reactions. We show here that for a large 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(u,v), and we discuss here a linear example which strongly hints at such behaviour.[PUBLICATION ABSTRACT]