Global blow-up of separable solutions of the vorticity equation

In this paper we construct solutions to the equation (*) delta(3)u/delta t delta y(2) = epsilon delta(4)u/delta y(4) + delta(3)u/delta y(3) - delta u delta(2)u/delta y delta y(2), epsilon > 0 on a finite interval in y which blow-up globally in finite time. This equation arises in a number of physical situations and can be derived from the vorticity equation by looking for stagnation-point type separable solutions for the two-dimensional streamfunction of the form xu(y, t). In the particular application which has prompted the investigation reported in this paper, (*) is solved subject to boundary conditions involving delta(2)u/delta y(2). For this type of boundary condition the phenomenon of blow-up was first observed numerically by solving the initial-boundary-value problem for (*). These computations reveal that, depending on the parameter combinations chosen, the solution to the initial-value problem may either blow-up globally in finite time or approach a steady state as t --> infinity. Using the computations as a guide we construct the analytic behaviour of the solution close to the blow-up time using the methods of formal asymptotics.