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 δ ∈ (0,1), we prove that there exist Schwartz class solutions $u(x,t)$ on R × [0,T) such that |u(x,t)| ≤ C 1+x²(x-x₀)² and u(x₀,t) (T-t)-δ as t→ T. We also prove that for any q∈ N, $T>0$, x₀∈ R, δ ∈ (0,1/2), there exist Schwartz class solutions $u(x,t)$ on R × [0,T) such that (i) |∂ₓ^q₁∂ₜ^q₂u(x,t)|≤ C for each q₁,q₂∈ N such that q₁+2q₂≤ q, (ii) |∂ₓ^q₁∂ₜ^q₂u(x,t)| ≤ C 1+|x||x-x₀| for each q₁,q₂∈ N such that q₁+2q₂= q+1, (iii) |∂ₓ^q₁∂ₜ^q₂u(x₀,t)| (T-t)-δ 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.