Nonlinear diffusion (heat) equation

Good day to all!
Let me consider equation
$$frac{partial u}{partial t}=frac{partial }{partial x}left(f(u)frac{partial u}{partial x}right)+g(u)$$
where $tin(0,+infty),xin [a,b]$ and $f(u),g(u)$ are smooth bounded functions.

In addition, given:

initial conditions
$$u(x,0)=varphi(x)$$

Neumann boundary condition
$$frac{partial u}{partial x}bigg|_{x=a}=frac{partial u}{partial x}bigg|_{x=b}=0,$$
I have a few questions:

1. What numerical method can be chosen to solve this problem? Can we be sure that the chosen method will be convergent and stable for this equation.

2. What can we say about Existence, Uniqueness and Stability of solution?

3. Can we transfer the answers of 2) to more general case

$$frac{partial bf{u}}{partial t}=text{div}(f(mathbf{ u})cdot text{grad},mathbf{u})+g(mathbf{u}),,,, text{where }mathbf{u}=(u_1(x,y,t),u_2(x,y,t),u_3(x,y,t))$$
$$mathbf{u}(x,y,0)bigg|_Omega=mathbf{Phi}(x,y),,,,,frac{partial mathbf{u}}{partial mathbf{n}}bigg|_Omega=0?$$