Validity of solution to PDE $xu_x - uu_t = t$ obtained from characteristics

Question

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:

$$ u(x, t) = \frac{\cos(\ln x) + \sin(\ln x)}{\cos(\ln x)-\sin(\ln x)}t $$

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.

Answer 1

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

• $\frac{\text d x}{\text d s} = x$. Letting $x(0)=1$ we know $x(s) = e^s$.
• $\frac{\text d t}{\text d s} = -u$ and $\frac{\text d u}{\text d s} = t$. Letting $t(0)=t_0$ and $u(0)=t_0$, we know $t(s) = t_0 \left(\cos s - \sin s\right)$ and $u(s) = t_0 \left(\cos s + \sin s\right)$.

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 $t = t_0 \left(\cos\ln x - \sin\ln x\right)$ where $t_0 \in \Bbb R$, and they carry the information $u = t_0 \left(\cos\ln x + \sin\ln x\right)$. All these curves intersect at the roots of the function $x \mapsto \cos\ln x - \sin\ln x$, i.e., at the abscissas $x_n = e^{\pi/4 + n\pi}$ with $n\in \Bbb Z$. At the abscissas $x_n$, the characteristic curves carry the value $u = \pm t_0\sqrt{2}$. 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 $x_0 = e^{5\pi/4} \approx 2.19$, and we can decrease $x$ until $x_{-1} = e^{-3\pi/4} \approx 0.095$ only. The classical solution is only valid for $s$ within the interval $]{-3\pi}/{4}, {5\pi}/{4}[$, 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 $s < {-3\pi}/{4}$ or $s > {5\pi}/{4}$.