Transition to Blow-up in a Reaction-Diffusion Model with Localized Spike Solutions

For certain singularly perturbed two-component reaction-diffusion (RD) systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behavior in terms of some parameter β in the system. For some such systems, such as the Gray-Scott model, a spike self-replication behavior is observed as a parameter varies across the saddle-node point. We demonstrate and analyze a qualitatively new type of transition as a parameter is slowly decreased below the saddle node value, which is characterized by a finite-time blow-up of the spike solution. More specifically, we use a blend of asymptotic analysis, linear stability theory, and full numerical computations to analyze a wide variety of dynamical instabilities, and ultimately a finite-time blow-up behavior, for localized spike solutions that occur as a parameter β is slowly ramped in time below various linear stability and existence thresholds associated with steady-state spike solutions. The transition or route to an ultimate finite-time blow-up can include spike nucleation, spike annihilation, or spike amplitude oscillation, depending on the specific parameter regime. Our detailed analysis of the existence and linear stability of multi-spike patterns, through the analysis of an explicitly solvable nonlocal eigenvalue problem, provides a theoretical guide for predicting which transition will be realized. Finally, we analyze the blow-up profile for a shadow limit of the RD system. For the resulting nonlocal scalar parabolic problem, we derive an explicit formula for the blow-up rate near the parameter range where blow-up is predicted. This blow-up rate is confirmed with full numerical simulations.