Avoiding instability and missing boundary condition errors when solving a system of 3 differential equations

I am trying to solve this system of equations:

dw(v,t)/dt =2g(v,t) w(t,v)
g(t,v)= Pi/2 (v^2)d g0(v,t)/dt
d go(v,t)/dt + d/dv[dw/dt *1/v^3)]=0

with initial conditions:

f0= (a/(Pi)^0.5)*e^[v^2/ve^2] + (b/(Pi)^0.5)*e^[(v - u)^2/ve^2]
a^2 + b^2 = 1
f0=go(v,0)
w(v,0)=0

This is my code

Here I am initializing some variables and defining f0 that is g(v,t=0).

a = 0.5;
b = (3^0.5)/2;
ve = 1;
u = 1;
f0[v_] = (a/(Pi)^0.5)Exp[v^2/ve^2] + (b/(Pi)^0.5)Exp[(v - u)^2/ve^2];
u1 = Log[((1 - a^2)/a^2)^0.5]/(2*u) + u/2;
u2 = 3u/2 - Log[((1 - a^2)/a^2)^0.5]/(2u);
f0[u1]
f0[u2]

Here I am using a system of 3 eqs for 3 functions and equating g(v,0)=f0

sol = NDSolve[{
D[go[v, t], t] + D[D[w[v, t], t], v]/(v^3) + (-3v^(-4))D[w[v, t], t] == 0,
g[v, t] == Pi/2 (v^2) D[go[v, t], v],
go[v, 0] == f0[v],
D[w[v, t], t] == 2g[v, t]w[v, t],
w[v, 0] == 0},
{go[v, t], W[v, t]},
{v, u1, u2},
{t, 0, 10000000000}]

And it is returning errors like

The PDE is convection dominated and the result may not be stable. Adding artificial diffusion may help

No DirichletCondition or Robin-type NeumannValue was specified for {g}; the result may not be unique

• The code contains several errors. There are no boundary conditions, which probably is the source of the second warning message. The limits of integraton in v are not specified. One of the variables in the code probably should be go instead of g. Note also that the second PDE can be solved symbolically up to a function of t only, which may allow the other two equations to be simplified. Dec 27 '21 at 4:05
• The limits of integration of v are specified. They are u1 and u2. u1 = Log[((1 - a^2)/a^2)^0.5]/(2*u) + u/2; u2 = 3u/2 - Log[((1 - a^2)/a^2)^0.5]/(2u); and go is a variable in the code... Dec 27 '21 at 5:40
• Cross-posted: community.wolfram.com/groups/-/m/t/2432445 Dec 28 '21 at 3:00