This is a problem appeared at Mathematica.SE. The OP wants to solve the following initial-boundary value problem of transport equation:
$$\frac{\partial u(x,t)}{\partial x}=\frac{\partial u(x,t)}{\partial t}$$ $$u(x,0)=x^2,u(0,t)=t$$
It can be easiliy shown the problem doesn't have a classical solution satisfying the i.c. and b.c. simultaneously, but one can still find a "solution" with the help of Laplace transform:
(* Corresponding _Mathematica_ code *)
{eq, ic, bc} = {D[u[x, t], x] == D[u[x, t], t], u[x, 0] == x^2,
u[0, t] == t};
teqn = LaplaceTransform[{eq, bc}, t, s] /. HoldPattern@LaplaceTransform[a_, __] :> a /.
Rule @@ ic
tsol = u[x, t] /. First@DSolve[teqn, u[x, t], x]
sol[x_, t_] = InverseLaplaceTransform[tsol, s, t]
$$u(x,t)=\left(-t^2-2 t x+t-x^2+x\right) \theta (t+x)+(t+x)^2$$
where $\theta$ is Heaviside step function. Further check shows this solution satisfies the equation except for $t+x=0$, the b.c. except for $t<0$ and the i.c. except for $x>0$.
So my question is, what have I found with Laplace transform? Is this the so-called weak solution? Well, to be honest, due to my poor math basis, I can't understand the description on the wikipedia page of weak solution very well, but $e^{-st}$ seems not to be a function with "compact support"?