I'll star by saying that I am not really familiar with the field of PDEs so this questions may be trivial or ill-possed in that case please let me know.


I am in search of some existence (Global) result regarding a system of first order PDEs with many state-variables and a non-homogeneity that is non-linear in the solutions, i.e.


\begin{align}
\begin{cases}
\partial\_t \mathbf u+\sum\_{n=1}^{K} A\_n(t)\partial\_{x\_n} \mathbf u=\mathbf b(t,\mathbf u,x\_1,\cdots,x\_K)\\
\mathbf u(0,x\_1,\cdots,x\_K)=u\_0\in\mathbb R^N,
\end{cases}
\end{align}


where the matrices $A\_n, n=1,\cdots, K$ are diagonal, and $\mathbf b$ is Lipschitz continuous in $\mathbf u$ uniformly in $t,x\_1,\cdots,x\_n$.


In Bressan's "Conservations laws" book the case with two variables is considered, namely $t$ and $x$. I have found some articles online considering the case with many state variables but in that case the non-homogeneity was linear in the $\mathbf u$ .


As far as I know the method of the characteristics applies only the one-dimensional case, so I am a little bit lost.


Is there some result I can use to show the existence (and uniqueness if possible) of a solution for this system?


Thanks in advance.



Existence of a solution for a quasilinear hyperbolic system of PDEs with many state variables
Okay, so I would write your equations instead in the following form:
$$ \partial\_t u\_i + v\_i(t) \cdot \nabla u\_i = b\_i (t, \vec{x}, \vec{u}) $$
This is a system of transport equations and so can actually be solved by using a variation of the Picard-Lindelof argument.


(I am implicitly assuming that your function $b$ is suitably nice in certain ways, which are implied by what you said is allowed in [this comment](https://mathoverflow.net/questions/377821/existence-of-a-solution-for-a-quasilinear-hyperbolic-system-of-pdes-with-many-st?noredirect=1#comment958928_377821).)


1
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Consider the iteration scheme
$$ \vec{w} \mapsto \vec{w}' $$
where $\vec{v}'$ solves the linear inhomogeneous equation
$$ \partial\_t w'\_i + v\_i(t) \cdot \nabla w'\_i = b\_i(t,\vec{x}, \vec{w})$$
with the *fixed*, given, initial data $\mathring{\vec{w}}:\mathbb{R}^K\to \mathbb{R}^N$.


Given a continuous function $\vec{w}: [-t\_0, t\_0] \times\mathbb{R}^K \to \mathbb{R}^N$, the function $\vec{w}'$ can be solved component wise by integrating along the integral curves of $\partial\_t + v\_i(t) \cdot \nabla$ on $[-t\_0, t\_0]\times \mathbb{R}^K$.


It thus suffices to show that this iteration scheme converges.


2
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Suppose $\vec{z}$ and $\vec{w}$ are given functions that are uniformly bounded by some constant $M$. Then there exists (under what I hope are reasonable assumptions on $b\_i$) some constant $M'$ such that $b\_i(t,\vec{x},\vec{z})$ and $b\_i(t,\vec{x},\vec{w})$ are uniformly bounded by $M'$. Thus you have that


$$ | \vec{z}'|\_\infty, |\vec{w}'|\_\infty \leq |\mathring{\vec{w}}|\_\infty + M' t\_0 $$


Choose $M = 2 |\mathring{\vec{w}}|\_\infty$ and there exists some $t\_0 > 0$ such that the iteration mapping maps the (closed) ball of radius $M$ in $C^0([-t\_0, t\_0]\times \mathbb{R}^K,\mathbb{R}^N)$ to itself.


3
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For the differences you find that
$$ \partial\_t (z'\_i - w'\_i) + v\_i(t)\cdot \nabla (z'\_i - w'\_i) = b\_i(t,\vec{x}, \vec{z}) - b\_i(t,\vec{x},\vec{w}) $$
Using the Lipschitz continuity in the final slot of $b\_i$ you get that
$$ \big|\partial\_t (z'\_i - w'\_i) + v\_i(t)\cdot \nabla (z'\_i - w'\_i) \big| \leq M'' |\vec{z} - \vec{w}| $$
for some $M''$. Integrating you get that if $t\_0$ is chosen sufficiently small, the iteration mapping is a contraction mapping.


4
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Applying Banach fixed-point theorem you get local existence.


5
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Having proven local existence, you can upgrade this to global in the same way you argue for ODEs: by showing that the sup norm of the functions involved does not blow-up. This follows from the *global* Lipschitz property of the functions $b\_i$, which guarantees that $|b\_i(t,\vec{x},\vec{w})| \leq M''' (1 + |\vec{w}|)$


So a standard Gronwall's inequality argument well tell you that the sup-norm of the solution cannot grow faster than $A e^{2 M''' |t|}$ for some constant $A$.


