Blow-up solutions of the "bad" Boussinesq equation

We study blow-up solutions of the ``bad" Boussinesq equation, and prove that a wide range of asymptotic scenarios can happen. For example, for each T>0, x₀ R and (0, 1), we prove that there exist Schwartz class solutions u (x, t) on R [0, T) such that |u (x, t) | C 1+x^2 (x-x₀) ^{2} and u (x₀, t) (T-t) ^- as t T. We also prove that for any q N, T>0, x₀ R, (0, 12), there exist Schwartz class solutions u (x, t) on R [0, T) such that (i) |ₗ^q₁ₓ^q₂u (x, t) | C for each q₁, q₂ N such that q₁+2q₂ q, (ii) |ₗ^q₁ₓ^q₂u (x, t) | C 1+|x||x-x₀| for each q₁, q₂ N such that q₁+2q₂= q+1, (iii) |ₗ^q₁ₓ^q₂u (x₀, t) | (T-t) ^- as t T for each q₁, q₂ N such that q₁+2q₂= q+1. In particular, when q=0, this result establishes the existence of wave-breaking solutions, i. e. solutions that remain bounded but whose x-derivative blows up in finite time.