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 $T>0$, $x_{0}\in \mathbb{R}$ and $\delta \in (0,1)$, we prove that there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R} \times [0,T)$ such that $|u(x,t)| \leq C \frac{1+x^{2}}{(x-x_{0})^{2}}$ and $u(x_{0},t)\asymp (T-t)^{-\delta}$ as $t\to T$. We also prove that for any $q\in \mathbb{N}$, $T>0$, $x_{0}\in \mathbb{R}$, $\delta \in (0,\frac{1}{2})$, there exist Schwartz class solutions $u(x,t)$ on $\mathbb{R} \times [0,T)$ such that (i) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x,t)|\leq C$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}\leq q$, (ii) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x,t)| \leq C \frac{1+|x|}{|x-x_{0}|}$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}= q+1$, (iii) $|\partial_{x}^{q_{1}}\partial_{t}^{q_{2}}u(x_{0},t)| \asymp (T-t)^{-\delta}$ as $t\to T$ for each $q_{1},q_{2}\in \mathbb{N}$ such that $q_{1}+2q_{2}= q+1$. 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.