LARGE TIME ASYMPTOTICS OF SOLUTIONS TO A MODEL COMBUSTION SYSTEM WITH CRITICAL NONLINEARITY

We consider a model semilinear reaction-diffusion system with cubic nonlinear reaction terms and small spacially decaying initial data on R(1). The model system is motivated by the thermal-diffusive system in combustion, and it reduces to a scalar reaction-diffusion equation with Zeldovich nonlinearity when the Lewis number is one and proper initial data are prescribed. For scalar equations of similar type it is well known that while a nonlinearity of degree greater than three (supercritical case) has no effect for large times a cubic nonlinearity qualitatively changes the long time behaviour. The latter case has been treated in the literature by a rescaling method under the additional assumption of smallness of the nonlinearity. Although for our system the cubic nonlinearity is also critical we establish large time behaviour when the nonlinearity is not necessarily small which essentially differs from the supercritical case. This is possible because of the interaction between nonlinear terms which have different signs. We show that although the system admits no self-similar solutions there do exist self-similar super (sub) solutions with new scaling exponents. This allows us to use a renormalization group analysis combined with a maximum principle.