Problem: consider the following PDE: $$-u_t=\mbox{sign}(u) u_x+ \frac{1}{2}u_{xx},$$ with some boundary condition $u(T,x)=\delta_a(x)-\delta_{-a}(x)$, $a>0$ fixed, being $\mbox{sign}(u)\in \{-1,1\}$ the sign of $u$ and $\delta$ the Dirac-function. The final condition could also be taken general $u(T,x)=\Phi(x)$ with $\Phi$ regular if it helps.
Here, $u$ is a function $u:[0,T]\times \mathbb{R}\rightarrow \mathbb{R}$.
Observe that if we change $\mbox{sign}(u)$ by simply a function $f(x)$ of $x$, this would make things easier or even if $f$ was constant, a semi-explicit solution using series could be found. However, $\mbox{sign}(u)$ is in some sense "simple" since it just changes sign from -1 to 1 and so on.
Question: Could there be a chance to find a (semi-explicit) solution or at least, properties of the sign of $u$? i.e. the regions where $u$ is positive, negative, etc. Is there any trick one could use for this specific kind of PDE?
Thanks for any tips or ideas!
Green's function generally is defined in the corresponding domain only. So what I meant can be written as $u(t,x)=G^+(x,a,t)1_{\{x>0\}}-G^-(-x,a,t)1_{\{x<0\}}$, where $G^+(x,a,t)$ is the solution of $−u_t=u_x+1/2u_{xx}$, $x>0$ and $u(T,x)=δ_a$, $u|_{x=0}=0$; $G^−(x,a,t)$ is the solution of $−u_t=-u_x+1/2u_{xx}$ and $u(T,x)=δ_a$, $u|_{x=0}=0$.
The solution $u$ and its derivatives $u_x$ and $u_t$ are continuous for $t<T$ but, as I've mentioned in the comments, $u_{xx}$ is not continuous on the line $x=0$.