I am trying to solve the PDE $$u_t(x,t)-ku_{xx}(x,t)=S_0\delta(x)\delta(t)$$ subject to the initial condition $u(x,0)=\delta(x)$.
Performing a Fourier transform on the PDE with respect to $x$ $(x\rightarrow w)$ which produces $$\frac{\partial}{\partial t}\hat{u}(w,t)+kw^2\hat{u}(w,t)=\frac{\delta(t)S_0}{\sqrt{2\pi}}.$$
We can then take a Laplace transform of this ODE with respect to $t$ $(t\rightarrow s)$ which produces $$s\hat{\bar{u}}(w,s)+kw^2\hat{\bar{u}}(w,s)=\frac{S_0}{\sqrt{2\pi}}+1, \tag{1}$$ as $\mathcal{F}_x(u(x,0))=\mathcal{F}_x(\delta(x))\implies \hat{u}(w,0)=\frac{1}{\sqrt{2\pi}}$ (using the definition with a factor of $\frac{1}{\sqrt{2\pi}}$).
My question is, is it even possible to invert $(1)$? This is a necessary step to solving the PDE.
Note that this post is similar to Problem with Heat Equation and Laplace Transform
Edit:
The $+1$ on the RHS appears as $$\mathcal{L}_t\left(\frac{\partial}{\partial t}\hat{u}(w,t)\right)=s\mathcal{L}_t(\hat{u}(w,s))-\hat{u}(w,0)$$