In the book J. Rauch, Partial Differential Equations, the author claims that for $\alpha\in(1/2,1)$, the function $u$ defined by $$u(x,t)=\int_{-\infty}^{\infty}e^{-(i\tau)^{\alpha}}e^{x(-i\tau)^{1/2}}e^{it\tau}d\tau,$$ where $z^{\alpha}$ and $z^{1/2}$ are understood as the principal branch, defines a solution of the heat equation $u_{t}-u_{xx}=0$ which is $C^{\infty}(\mathbb{R}^{2})$, not identically zero, $u(x,t)\equiv 0$ for $t\leq 0$, and "grow exponentially fast as $\left|x\right|\rightarrow\infty$ and are not tempered distributions in $x$." (see Section 3.9, bottom of pg. 125).
I'm having trouble rigorously showing this last assertion. I see that I can write the integrand as $$\exp\left(-\cos(\text{sgn}(\tau)\frac{\pi}{2}\alpha)\left|\tau\right|^{\alpha}+x\left|\tau\right|^{1/2}\cos(\text{sgn}(\tau)\frac{\pi}{4})\right)\exp\left(i\left[-\sin(\text{sgn}(\tau)\frac{\pi}{2}\alpha)\left|\tau\right|^{\alpha}-x\sin(\text{sgn}(\tau)\frac{\pi}{4})\left|\tau\right|^{1/2}+t\tau\right]\right)$$ So the modulus of the integrand is growing exponentially in $x$. But I'm not sure how to use this observation when I take the integral. Note that it's not necessarily true that a null solution to the heat equation is spatially unbounded for fixed $t>0$. So a proof of the author's claim will require asymptotic analysis of this particular solution.
