Heat Conduction in a Finite Bar with a Nonlinear Source

A model reaction-diffusion equation for temperature with a nonlinear source term is introduced which is an extension of the linear source one treated in Chapter 5 . This is equivalent to the interaction-diffusion equation for population density originally analyzed by Wollkind et al. (SIAM Rev 36:176–214, 1994, [142]) through terms of third-order in its supercritical parameter range. That analysis is extended through terms of fifth-order to examine the behavior in its subcritical regime. It is shown that under the proper conditions the two subcritical cases behave in exactly the same manner as the two supercritical ones unlike the outcome for the truncated system. Further there also exists a region of metastability allowing for the possibility of population outbreaks discussed in Chapter . These results are then used to offer an explanation for the occurrence of isolated vegetative patches and sparse homogeneous distributions in the relevant ecological parameter range where there is subcriticality for a plant-ground water model to be treated in Chapter . Finally these results are discussed in the context of Matkowsky’s (Bull Amer Math Soc 76:646–649, 1970, [78]) two-time nonlinear stability analysis. The problem applies this nonlinear stability analysis through terms of third-order to a particular reaction-long range diffusion model equation (Wollkind et al, SIAM Rev 36:176–214, 1994, [142]).