Is it implied that the problem is well-defined?

[evinda]

Hello! (Wave)

We have the Cauchy problem of the equation

$u_t+xu_x=xu, x \in \mathbb{R}, 0<t<\infty$

with some given smooth ($C^1$) function $g$ as initial value.

I want to check if the problem is well defined for each time. We know that a problem is well defined if the solution exists, is unique and depends continuously on the data of the problem.

I have computed that the solution of the problem is $u(x,t)=g(xe^{-t}) e^{x(1-e^{-t})}$.

So we have that the problem is well-defined if $g$ does not take two different values for some specific $t$, right?

But is this implied from the fact that $g$ is smooth? (Thinking)

 

[S.G. Janssens]

It it implied from the fact that $g$ is a function: For each argument of $g$ (i.e. $x e^{-t}$) there exists precisely one function value of $g$.

So, you have found a (global, classical) solution $u \in C^1(\mathbb{R} \times (0,\infty))$.
Now you would need to check that it is unique, and you would need to think what it means for $u$ to depend continuously on the initial datum $g$.

 

[evinda]

It it implied from the fact that $g$ is a function: For each argument of $g$ (i.e. $x e^{-t}$) there exists precisely one function value of $g$.

So, you have found a (global, classical) solution $u \in C^1(\mathbb{R} \times (0,\infty))$.
Now you would need to check that it is unique, and you would need to think what it means for $u$ to depend continuously on the initial datum $g$.

How could we show the uniqueness of the solution? Do we suppose that there is an other solution? (Thinking)

Isn't it implied directly that $u$ depends on the initial data, since it contains $g$ ?

 

[I like Serena]

How could we show the uniqueness of the solution? Do we suppose that there is an other solution? (Thinking)

Isn't it implied directly that $u$ depends on the initial data, since it contains $g$ ?

As $g$ is smooth of class $C^1$, that means that the solution is well-defined for any $x$ and for any $t$.

And since the solution contains $g$ as the initial value for (presumably) $t=0$, doesn't that imply that the solution is unique?
After all, a different instance of $g$ means that we have a different solution, don't we? (Wondering)

 

[evinda]

And since the solution contains $g$ as the initial value for (presumably) $t=0$, doesn't that imply that the solution is unique?

So because of the fact that $g$ is well-defined for any $x$ and $t$, and $u$ contains $g$, we get that the solution $u$ is unique? (Thinking)

After all, a different instance of $g$ means that we have a different solution, don't we? (Wondering)

Yes, when we have a different $g$ we get a different solution, so $u$ depends continuously on the data of the problem, right? (Thinking)

 

[S.G. Janssens]

So because of the fact that $g$ is well-defined for any $x$ and $t$, and $u$ contains $g$, we get that the solution $u$ is unique? (Thinking)

I don't think the uniqueness of the solution follows like that.

It seems to me it is best to go back and consider how you obtained your solution. Was it with the method of characteristics? If so, then what does the method of characteristics tell you about existence and uniqueness of global (that is, defined for all $t > 0$) solutions for this problem?

Yes, when we have a different $g$ we get a different solution, so $u$ depends continuously on the data of the problem, right? (Thinking)

Of course $u$ depends on $g$, but continuous dependence is another matter. Think of the map that sends $g$ to $u$:
\[
g \mapsto u(\cdot,\cdot, g) \qquad (*)
\]
where the dots are place-holders for $x$ and $t$. So, this map sends an initial condition $g$ to the solution $u(\cdot,\cdot,g)$, which is a function of the two variables $x$ and $t$. As the domain of $(*)$ we could take $C^1(\mathbb{R})$ and as its co-domain $C^1(\mathbb{R} \times [0,\infty))$.

Now the question of continuous dependence on initial conditions becomes a question of continuity of this map, but continuity with respect to what? We need a way to measure the changes in $u(\cdot,\cdot,g)$ as a result in changes in $g$. (Put more fancily: We need a topology on the domain and co-domain of (*).)

I can think of many ways to do that, typically involving $C^1$ function norms of $g$ and $u$ and compact subsets of $\mathbb{R}$ and $[0,\infty)$, but it may be better to ask you whether in your lectures you were provided with any more precise definition of "continuous dependence" for this kind of problem. It is this definition that you should probably recall, formulate and apply.

(Ultimately, this application is probably not going to be difficult, because you have an explicit solution that depends on $g$ in a nice way.)