INTERIOR BLOWUP IN A CONVECTION-DIFFUSION EQUATION

This paper addresses the qualitative behavior of a nonlinear convectio n-diffusion equation on a smooth bounded domain in R-n, in which the s trength of the convection grows superlinearly as the density increases . While the initial-boundary value problem is guaranteed to have a loc al-in-time solution for smooth initial data, it is possible for this s olution to be extinguished in finite time. We demonstrate that the way this may occur is through finite-time ''blow up,'' i.e., the unbounde dness of the solution in arbitrarily small neighborhoods of one or mor e points in the closure of the spatial domain. In special circumstance s, such as the presence of radial symmetry, the set of blowup points c an be identified; these points may be either on the boundary or on the interior of the domain. Furthermore, criteria can be established that guarantee that blowup occurs. In this paper, such criteria are presen ted, involving the dimension of the space, the growth rate of the nonl inearity, the strength of the imposed convection field, the diameter o f the domain, and the mass of the initial data. Furthermore, the tempo ral rate of blowup is estimated.