List of top Partial Differential Equations Questions on Heat Equation and Fourier Transform Methods

Let \( u(x,t) \) be the solution of the initial value problem for the heat equation on the real line:
\[ \frac{\partial u}{\partial t} = k\, \frac{\partial^2 u}{\partial x^2}, \qquad -\infty< x< \infty, \qquad t>0, \qquad k\in\mathbb{R} \]
with the initial condition
\[ u(x,0) = e^{-a|x|}, \qquad a>0. \]
If \( \hat u(w,t) = \displaystyle\int_{-\infty}^{\infty} u(x,t)\, e^{iwx}\, dx \) is the Fourier transform of \( u(x,t) \) with respect to \( x \), then \( \hat u(w,t) \) is equal to