Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$:
Given a function $u(x,t) \in L^2([0,T];H^2(\mathbb{R}))$, and $\alpha(t), \beta(t)$ which are injective and continuous, we have that $$v(x,t) = \alpha(t)u(\alpha(t)x, \beta(t)).$$
At this point, I am struggling to understand how to justify that $v(x,t)$ is in $L^2([0,T];H^2(\mathbb{R}))$. What must $\alpha(t), \beta(t)$ satisfy? Or just $\alpha(t)$, since it is the only one "outside" which I would need to control for when taking the norm?
In my case, $\alpha(t)=e^t$ and $\beta(t)=\frac{e^{5t}-1}{5}$.