You’ve overwritten the variable u such that when you compute the Frechet derivative necessary for Newton’s method in solve(F == 0, u) you get a zero Jacobian since F does not contain your new u.
If you delete the line
...
# Time-stepping
u = Function(V) # Delete this
...
you will at least have a system which can be assembled.
Futhermore, for some hints and tips regarding nonlinear problems, see here.