Dgdrxrt

I have a boundary-value problem:

$$
xu_x - uu_t = t
$$

with boundary conditions:

$$
u(1, t)= t, -infty<t<infty
$$

Finding the characteristic equations is no problem, and I get a general solution of:

$$
u(x, t) = fraccos(ln x) + sin(ln x)cos(ln x)-sin(ln x)t
$$

I have base characteristics that look like this:

And I know that my solution is defined only when $x>0$.

However, I am having trouble gaining any intuition on how to interpret these characteristic plots, and determining well-posedness.

So my main questions are:

1. For this example, is the initial curve the line $t=0$?




2. Is the solution defined everywhere for $x>0$? I would say yes.




3. Is the solution unique? I would say no, because of the intersection of all of the base characteristics at that point on the $x$-axis implies that I can have the same solution for different characteristics.




4. Are there restrictions on my parameters? For example can I allow my parameters to run over arbitrary intervals?




5. Although I know the criteria for a well-posed problem, I feel like I still cannot determine whether this problem is well posed. I would say no, because it fails uniqueness.




6. Is the solution single valued everywhere? I have no idea.

For this problem, the boundary data is located along the line $x=1$. Let us apply the method of characteristics:

• 
  $fractext d xtext d s = x$. Letting $x(0)=1$ we know $x(s) = e^s$.


• 
  $fractext d ttext d s = -u$ and $fractext d utext d s = t$. Letting $t(0)=t_0$ and $u(0)=t_0$, we know $t(s) = t_0 left(cos s - sin sright)$ and $u(s) = t_0 left(cos s + sin sright)$.

The expression of $x(s)$ gives $s = ln x$ for positive $x$. Combining the expressions of $t(s)$ and $x(s)$, one obtains indeed the expression of the solution $u(x,t)$ in OP.

Now, let us analyze the expression of characteristic curves, which tell how the boundary information located at $x=1$ propagates in the $x$-$t$ plane. These curves are expressed by the equations $t = t_0 left(cosln x - sinln xright)$ where $t_0 in Bbb R$, and they carry the information $u = t_0 left(cosln x + sinln xright)$. All these curves intersect at the roots of the function $x mapsto cosln x - sinln x$, i.e., at the abscissas $x_n = e^pi/4 + npi$ with $nin Bbb Z$. At the abscissas $x_n$, the characteristic curves carry the value $u = pm t_0sqrt2$. Therefore, the classical solution $u(x,t)$ deduced from the characteristics is multivalued at the abscissas $x_n$. Since the boundary-value problem sets the data at $x=1$, we can increase $x$ until the solution becomes multivalued at $x_0 = e^5pi/4 approx 2.19$, and we can decrease $x$ until $x_-1 = e^-3pi/4 approx 0.095$ only. The classical solution is only valid for $s$ within the interval $]-3pi/4, 5pi/4[$, such that $x$ belongs to $]x_-1, x_0[$.

The boundary-value problem is well-posed, but the solution blows up at some point. With the present boundary data located at $x=1$, it does not make sense to look for values of $s < -3pi/4$ or $s > 5pi/4$.