I am considering the PDE of the following type $$\frac{\partial W}{\partial u}=\frac{\partial^{2} W}{\partial x^{2}},$$ $ u>0,\:0<x<\infty$ such that $W(0,x)=\delta(x)$. I wonder if this problem is solvable and the corresponding solution if it is solvable.
Another PDE that I would like to consider is as follow: $$\frac{\partial W}{\partial u}=\frac{\partial^{2} W}{\partial x^{2}},$$ $ u>0,\:\theta(u)<x<\infty$ such that $W(0,x)=\delta(x)$, $\theta(0)=0<x<\infty$ and $W(u,\theta(u))=0$. This is a PDE of moving boundary. Is it possible to decompose $W=W^{1}+W^{2}$ such that $$\frac{\partial W^{1}}{\partial u}=\frac{\partial^{2} W^{1}}{\partial x^{2}},$$ $ u>0,\:-\infty<x<\infty$ such that $W^{1}(0,x)=\delta(x)$ and $$\frac{\partial W^{2}}{\partial u}=\frac{\partial^{2} W^{2}}{\partial x^{2}},$$ $ u>0,\:\theta(u)<x<\infty$ such that $W^{2}(u,\theta(u))=-W^{1}(u,\theta(u))$, $W^{2}(0,x)=0$?