I have a function of three variables, and I'm wondering if the method of characteristics can be used to solve the PDE. Specifically, let $u(t, x, y)$ be a function of three variables satisfying the following PDE: $$ \frac{\partial u}{\partial t}= f(x,y) \frac{\partial u}{\partial x}+g(x,y)\frac{\partial u}{\partial y}, $$ where $f(x,y)$ and $g(x,y)$ are given functions that do not depend on $t$.
Given boundary data and a characteristic curve $(t, x(t), y(t))$ such that $u(t,x(t),y(t))$ is constant, can an explicit solution for $u$ be found?
$$ \frac{\partial u}{\partial t}= f(x,y) \frac{\partial u}{\partial x}+g(x,y)\frac{\partial u}{\partial y}, $$ $$u_t-f(x,y)u_x-g(x,y)u_y=0$$ The Charpit-Lagrange characteristic ODEs are : $$\frac{dt}{1}=\frac{dx}{-f(x,y)}=\frac{dy}{-g(x,y)}=\frac{du}{0}$$ A first characteristic equation comes from solving $\frac{dx}{-f(x,y)}=\frac{dy}{-g(x,y)}$ $$\frac{dy}{dx}-\frac{g(x,y)}{f(x,y)}=0$$ Without knowing explicitly the functions $f(x,y)$ and $g(x,y)$ it isn't possible to say if this ODE is analytically solvable or not.
Supposing that the solution of the ODE can be found on the form : $$\varphi(x,y)=c_1$$ with $\varphi$ a function explicitly known, then one could continue.
A second characteristic equation comes from solving $\frac{dt}{1}=\frac{dx}{-f(x,y(x,c_1))}$
$y(x,c_1)$ supposes that one can solve $\varphi(x,y)=c_1$ for $y$. $$t+\int \frac{dx}{f(x,y(x,c_1))}dx=c_2$$
Supposing that one can explicitly integrate, then the second caracteristic equation would be : $$t+\phi(x,c_1)=t+\phi(x,\varphi(x,y))=c_2$$ with $\phi$ a function explicitly known.
If all the above calculus can be carried out, the general solution of the PDE would be $u=F(c_1,c_2)$ : $$u(t,x,y)=F\big(\varphi(x,y)\:,\:(t+\phi(x,\varphi(x,y)))\big)$$ in which $\varphi$ and $\phi$ are known functions and in which $F$ is an arbitrary function of two variables.
If some valid conditions (initial and/or boundary) are specified, one expect to determine the function $F$ in order to satisfy those conditions.
Thus the answer to your question about the possibility to solve the PDE thanks to the method of characteristics is : It is theoretically possible but not explicitly for all kind of functions $f(x,y)$ and $g(x,y)$.
It is not possible to definitively answer without knowing explicitly what are the functions $f(x,y)$ and $g(x,y)$.