Let $w(x_1, \dots, x_n)$ be a $2k$-degree polynomial of $n$-variables given by
$$w(\vec x) = \sum_{\alpha : |\alpha| \leq 2k} c_\alpha x^\alpha$$
where $c_\alpha \in \mathbb C$ and $\text{Re}(c_\alpha) \geq 0$.
The notation $w(-i \nabla)$ is somewhat idiosyncratic and means
$$w(-i \vec\nabla) = \sum_{\alpha : |\alpha| \leq 2k} c_\alpha (-i \partial)^\alpha$$
where $(-i\partial)^\alpha = \left((-i\partial_{x_1})^{\alpha_1},\dots,(-i\partial_{x_n})^{\alpha_n}\right)$.
I wish to solve $$-w(-i \vec \nabla) u = u_t$$ taking Fourier transforms assuming the necessary regularity of $u$, we have $$-\mathcal F \{w(-i \vec \nabla) u\} = \partial_t \mathcal F \{u\}$$ which simplifies to the following using Fourier multipliers $$\hat u _ t = -w(\vec \xi) \hat u$$ and $$\hat u (\xi, t) = Ce^{-w(\vec \xi)t}$$ In the above, $\vec \xi$ is the frequency space representation of $\vec x$. $$ \begin{align*} u(\vec x, t) &= C\int_{\mathbb R ^ n} \exp\left(i \vec \xi \cdot \vec x - w(\vec \xi)\right) d \vec \xi \\ &= C\int_{\mathbb R ^ n} \exp\left(i \vec \xi \cdot \vec x - \sum_{\alpha : |\alpha| \leq 2k} c_\alpha \xi^{\alpha} \right) d \vec \xi \end{align*} $$
Question: How can I compute the following integral when one term of the polynomial involves multiple $\xi$? When $w(x) = x^2$, we get our beloved heat kernel. Eventually, I would like to use $u(\vec x, t)$ to produce a kernel for solving a general Cauchy problem.
I was able to look up the following indefinite integral identity
$$ \int_{0}^\infty e^{-x^n}dx=\Gamma \left(1 + \frac{1}{n}\right) $$
With the necessary u substitutions, I think this would be sufficient to solve the case of any polynomial in a single variable. What I don't know is how to handle cross terms in a polynomial of multiple variables.
I know a solution exists, because the method of Fokas can resolve this question on the half line. Under the appropriate boundary conditions on $w$, the integral can be made to converge (potentially with loss of generality for the admissible $w$).
Bonus: Does anyone know of a relationship between the solution $u(\vec x,t)$ and an algebraic variety formed by $-w(\vec \xi)$? I was thinking that the stationary phase approximation from physics would state that the greatest contribution to $u(x,t)$ would occur when $w(\vec \xi) \approx 0$.