Interior Blowup in a Convection-diffusion Equation

This paper addresses the qualitative behavior of a nonlinear convection-diffusion equation on a smooth bounded domain in Rn, in which the strength of the convection grows superlinearly as the density increases. While the initial-boundary value problem is guaranteed to have a local-in-time solution for smooth initial data, it is possible for this solution to be extinguished in finite time. We demonstrate that the way this may occur is through finite-time “blow up”, i.e., the unboundedness of the solution in arbitrarily small neighborhoods of one or more points in the closure of the spatial domain. In special circumstances, such as the presence of radial symmetry, the set of blowup points can be identified; these points may be either on the boundary or the interior of the domain. Furthermore, criteria can be established that guarantee that blowup occurs. In this paper, such criteria are presented, involving the dimension of the space, the growth rate of the nonlinearity, the strength of the imposed convection field, the diameter of the domain, and the mass of the initial data. Furthermore, the temporal rate of blowup is estimated.