The viscous Burgers’ equation without diffusion is given by: $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0 $$ Now the solution should satisfy the implicit form: $u=f(x-ut)$. However as we keep increasing $t$, the implicit form no longer has an explicit form and hence the solution should not exist.
The numerical test suggest that upon applying upwind schemes for initial condition $f(x)=sin(x)$ on the periodic domain $[0,2\pi]$, the wave deforms towards the stationary point $x=\pi$ as expected (due to different velocity $u$ at different points). As we keep increasing $t$, due to dissipiative errors (most probably), the solution flattens out.
Now my question is, in reality how the solution will behave asymptotically? That is, how will be the solution if we keep increasing $t$? Would it reach a discontinuous function in finite time? In that case is it possible to derive that finite time? Or will it pointwise converge to a discontinuous function in infinite time?