First make sure you use double equal signs when forming an expression such as u(x,0,t)==0, also, use parentheses to group items in an equation not brackets. What I would do is start small and then scale up. So I started with the construct for a 1D heat equation, got that error free, then scaled up to a simple 2D heat equation with five boundary conditions and got that error free. Then I tried your equation with those same conditions but ran into a problem with the Norm so I just used a square root for the norm but I still specified five sides of the box as the boundary conditions and initial conditions using just the exponential function . If you like, you can start with my code below and see if you can gradually change it into the IBVP you desire.

Clear[u]

myEqn = D[u[x, y, t], t] - 1 - (D[u[x, y, t], {x, 2}] +

D[u[x, y, t], {y, 2}]) + Sqrt[D[u[x, y, t], x]^2 +

D[u[x, y, t], y]^2] == 0

sol = NDSolve[{myEqn, u[x, 0, t] == Exp[-x^2],

u[x, 1, t] == Exp[-(x^2 + 1)],

u[0, y, t] == Exp[-y^2],

u[1, y, t] == Exp[-(1 + y^2)],

u[x, y, 0] == Exp[-(x^2 + y^2)]},

u, {x, 0, 1}, {y, 0, 1}, {t, 0, 1}]