Blow up of solutions of a generalized Boussinesq equation

Abstract

Consider the Cauchy problem formula math where f R → R C∞, 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 α > 0 be real, and such that 2(1 + 2α)F(u) ≥ uf(u), (2) 2/2(υ 0 , Pυ 0 ) L 2 + ∫ - ∞∞ F(u 0 )dx < 0, (3) where P = 1-∂ 2 /∂x 2 , F'(s) = f(s), and υ 0 is given by u t (x, 0) = (υ 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.