DYNAMICS OF A NONLINEAR CONVECTION-DIFFUSION EQUATION IN MULTIDIMENSIONAL BOUNDED DOMAINS

The scalar nonlinear convection-diffusion equation u(t) - v Delta u a(u) . Vu = g(x), t > O, is considered, for given initial data and zer o Dirichlet boundary conditions, in a smooth bounded domain Omega subs et of R(n). The homogeneous viscous Burgers' equation in one dimension is well-known to possess a unique, exponentially attracting equilibri um. These properties are shown to be preserved in the generalisation c onsidered. Furthermore, the equilibrium is shown to be bounded in the maximum norm independently of the function a. The main methods used ar e maximum principles, and a variational method due to Stampacchia.