Is there a solution to this unidirectional wave equation, with initial value $v=f(x)$ and $x=t^2$

Question

unidirectional wace equation:

$$\frac{du}{dt}+c\frac{du}{dx}=0$$

The initial value $u=f(x)$ is given on the parabola $x=t^2$.

Is there a solution to this problem, discuss why the solution is unique and differentiable or discuss why there is no solution.

and is there a solution to the problem if $t\leq c/2$

Can someone help me how to start with this? Do I need Fredholm alternative to see how many solutions. Or what should be the first step. Characteristics maybe?

Answer 1

You have your initial condition on the curve defined by $x=t^2$. At the same time you should know that the characteristics to the transport equation are the strait lines $$ x=ct+\xi, $$ where $\xi$ is some constant, and your solution is constant along the characteristics.

Clearly, no matter what the value of $c$ the straight line with $\xi=0$ will cross the curve of the initial conditions twice, since we know that $f$ is arbitrary hence this will imply that in general on the same characteristic there will be two different initial conditions, hence no solution exists in general.

The case $t\leq c/2$ I will leave to you. You need to analyze your initial conditions and characteristics and conclude that...