Existence and uniqueness for a system of first-order PDE

Question

Let $y$ be a scalar and ${\bf t}=(t_1,\ldots,t_K)$ and \begin{align*} \frac{\partial y({\bf t})}{\partial t_k} &=f_k(y({\bf t}),{\bf t}) \qquad k=1,\ldots,K\\ y(t_{10},\ldots,t_{K0}) &=y_0 \end{align*} Assume $f_k$ are Lipschitz continuous. Does a solution exist? Is it unique? I think the answer is yes because fixing all but one coordinate this is an ODE and by Picard–Lindelöf theorem the solution exists and is unqiue.

Answer 1

The first-order PDEs can indeed be related to ODE by the method of characteristics. But this does not make them identical to PDE. A nod to Picard–Lindelöf is not nearly enough. Yes, you can solve an ODE along some curves - but when those curves intersect, do you get the same result?

Also, you have not a single PDE but a system of $K$ equations. Yet, there is just one scalar unknown. This means the system is overdetermined: a solution will only exist in exceptional cases.

Here is an illustration: $$\frac{dy}{dt_1} = t_2,\quad \frac{dy}{dt_2} = 0 $$ Does not get much simpler than that. All kinds of Lipschitz conditions are satisfied. Yet, there is no solution. Indeed, the second equation says that $y$ is independent of $t_2$. But then how can its $t_1$-derivative depend on $t_2$?

The above equation is of simple kind, where $f_k$ do not depend on $y$ at all. In this case, solution exists if and only if the field $(f_1,\dots,f_K)$ is conservative.