I am confused about different cases of boundary condition for half space heat equation problem and I describe each case next. The basic problem setup is to solve $$ \mathrm{u}_{t} = \mathrm{u}_{xx}\,,\qquad x > 0\,,\quad 0 < t \leq T $$ with initial data $\mathrm{u}\left(\, 0,x\,\right) = f(x)$, consider $f(x)$ to be continuous for all $x>0$ for the first two cases and $\mathrm{u}\left(\, t,0\,\right)=g(t)$.
- First is the case of constant $g(t)=0, f(0)\neq 0$, so we extend $f(x)$ in an odd way, i.e. for negative x $f(-x)=-f(x)$ and use the solution to the full problem to derive the solution for the half space. I can follow the mothod and see how to end with a solution. However, what happens in that corner $(0,0)$? If I plug zero into $f(-0)=-f(0)$ does it imply $f(0)=0$? And if this is the case, does it imply that $f(x)$ is continuous? So, the solution inside of the domain is extremely smooth, but then is that up to the boundary?
The case if $g(t)$ is some function, $f(0)\neq 0$, now the trick is to split the solution into the problem above and another one that has zero initial data and boundary condition $g(t)$. There is also an integral can be written as a solution to this second part of the problem and it blows up for $t=0,x=0$. So, then what, the total solution also is supposed to blow up? One can say initially I set $\tilde{U}=U-g(0),\tilde{f}=f-g(0),\tilde{g}=g-g(0)$, and rewrite my initial problem for all the new functions $\tilde{U},\tilde{f},{\tilde{g}}$, then all the problems go away and I have infinitely smooth solution everywhere $x\geq 0$ and up to the boundary?
Lastly, add on top of the problem two discontinuity in $f$, for example if it is a step function for some constant $x=c>0$. Is that yet another case to define the regularity for the solution?
I am confused about "corner" discontinuity? When is that an issue and when is it not? Does it depend on boundary condition $g(t)$? I don't intend to see any proofs, I am after intuition and just facts about those cases. I just want to know the end result for the above cases. (sorry I don't see why it compiles only part of the question)