Global existence of solutions to the Proudman-Johnson equation

We show that there is no blow-up solutions, for positive viscosity constant nu, to the equation f(xxt) - nuf(xxxx) + ff(xxx) - f(x)f(xx) = 0, x is an element of (0. 1).t > 0 with (i) periodic boundary condition, or (ii) Dirichlet boundary condition f = f(x) = 0 or (iii) Neumann boundary condition f = f(xx) = 0 on the boundary x = 0, 1. Furthermore we show that every solution decays to the trivial steady state as t goes to infinity.