Is there a PDE whose steady state would result in a discontinuity. Namely I know that with the advection equation by prescribing Dirichlet boundary conditions I can propagate those in the form of discontinuities. For example:
$$u_t + u_xx = 0, \, u(t,a) = y, \, u'(t,b) = 0, u(0,x) = 0$$
The above would propagate a discontinuity of height $y$ from left to right, and if I negate the sign and flip the boundary conditions I would get it from right to left.
Is there a PDE that would result in a propagation from both sides, and preserving the discontinuity between the two in the middle. That is given $u(t,a) = y_a, \, u(t,b) = y_b$, I need a PDE that will result in the solution $u(\infty,x) = y_a, \,x\in [a,(a+b)/2)$ and $u(\infty, x) = y_b, \, x \in ((a+b)/2, b]$ without me having to artificially prescribe the discontinuity.