Global Existence for Nonlinear Diffusion Equations

Abstract

Solutions of nonlinear (possibly degenerate) reaction-diffusion models are known to exist for all time if the asymptotic growth of the reaction term is not greater than that of the diffusion term. Via concavity methods, it is also known that negating such a condition results in solutions which blow up in finite time. On the other hand, two recent studies regarding the case of linear diffusion have reported on the ability for convective terms in a model to create global existence of solutions where no such result is true in their absence. In this paper, we develop a theory of global existence for a general class of reaction-diffusion-convection models which only requires a similar balance of diffusion and reaction. Our result extends previous work to models including convection, but it does not reveal the exact role of convection in yielding global solutions. Further analysis of a slightly simplified model establishes the global existence of all solutions if the reaction term grows asymptotically at a rate less than that of either diffusion or convection.