I'm just learning Mathematica and I'm having trouble with the PDE solver NDSolve... I use Freefem usually for PDEs but I just want a quick and dirty way of solving here.

The PDE is given by

sol = D[u[x, y, t], t] - 1 -

a*{D[u[x, y, t], x, x] + D[u[x, y, t], y, y]} -

b*Norm[{D[u[x, y, t], x], D[u[x, y, t], y]}] == 0

with a and b specified earlier.

Then we have the periodic conditions

cond = u[x+2*Pi,y+2*Pi,t] == u[x,y,t]

and the initial condition

cond2 = u[x,0,0] = 0

Then I should use

NDSolve[{sol,cond,cond2},u,{x,0,1},{y,0,1},{t,0,1}]

Unfortunately I get errors.

NDSolve:verdet: There are fewer dependent variables, {u[x,y,t]}, than equations, so the system is overdetermined. >>

but I've specified three equations. If I relax that IC and just use the periodic conditions I get the same error. If I remove the periodic conditions and just use the IC I get

NDSolve::bcedge: Boundary condition u[x,0,0]==0 is not specified on a single edge of the boundary of the computational domain. >>

Which I can't seem to fix.

So I'm bringing this to a community of people... I need your help! This is just my dummy problem for a larger class of PDES I'm trying to solve, so I really need to figure this out... and for the life of me I cant!

P.S.

I really want to use u(x,y,0) = 0 but this cannot be used because setting that as cond2 returns true, not an equation. Is there a way around this?