Blow up of solutions of a generalized Boussinesq equation

Consider the Cauchy problem [GRAPHICS] where f : R --> R C-infinity, f(0) = 0. After treatment of the local existence problem, we show the blow up of the solution of the equation (1) under the following assumptions. Let alpha greater-than 0 be real, and such that [GRAPHICS] where P = 1 - partial derivative(2)/partial derivative x(2), F'(s) = f(s), and nu(0) is given by u(t)(x, 0) = (upsilon(0)(X))(x). Then we focus on various perturbations of the equation. We also study the vectorial case in the same way, and finally we give some examples.