Solve nonlinear coupled PDE with x,y and t dependent

Question

I am trying solve the following two coupled PDE equations.

∂f/∂t=(∂^2 f)/(∂x^2 )+(∂^2 f)/(∂y^2 )-1/f^3 [(∂B/∂x)^2+(∂B/∂y)^2 ]-f+f^3

(∂^2 B)/(∂x^2 )+(∂^2 B)/(∂y^2 )-B*f^2+2/f [(∂B/∂x)(∂f/∂x)-(∂B/∂y)(∂f/∂y)]=0

The boundary conditions are

B(t,0,y)=const*t,B(t,x,0)=B(t,x,10),B(t,10,y)=0,

f(t,10,y)=1,f(t,x,0)=f(t,x,10),

Initial condition

f(0,x,y)=1 and B(0,x,y)=0

In order to solve I set up a square region {x=0,10 and y=0,10}.Left wall we don't know how f behaves.When time ramp up applied B increases so I need to know how B and f changes with time inside the box.

When I try to solve I get following errors:

1.NDSolve::pdord: Some of the functions have zero differential order, so the equations will be solved as a system of differential-algebraic equations.

2.NDSolve::bcart: Warning: an insufficient number of boundary conditions have been specified for the direction of independent variable x. Artificial boundary effects may be present in the solution.

I have following questions:

1.Is it possible to solve this type of coupled pde using ndsolve?

2.Is there any method I can use to over come this error? Please help..I am new to Mathematica.. Code is as follows.here u==f and v==B**

 h = 0; inf = 10; k = 3;
pde = {D[u[t, x, y], t] == 
    D[u[t, x, y], {x, 2}] + 
     D[u[t, x, y], {y, 
       2}] - (k^4/u[t, x, y])^3*(D[v[t, x, y], y]^2 + 
        D[v[t, x, y], x]^2) - u[t, x, y] + u[t, x, y]^3, 
       D[v[t, x, y], {x, 2}] + 
     D[v[t, x, y], {y, 2}] - (v[t, x, y]*u[t, x, y]^2)/k^2 + 
           (2/u[t, x, y])*(D[u[t, x, y], x]*D[v[t, x, y], x] - 
        D[u[t, x, y], y]*D[v[t, x, y], y]) == 0}; 
ic = {u[0, x, y] == 1, v[0, x, y] == 0};
bc = { u[t, inf, y] == 1, v[t, 0, y] == h + 0.1*t, 
   v[t, x, 0] == v[t, x, inf], u[t, x, 0] == u[t, x, inf], 
   v[t, inf, y] == 0}; 
NDSolve[{pde, bc, ic}, {u, v}, {x, 0, inf}, {y, 0, inf}, {t, 0, 15}];
Table[Plot3D[
  Evaluate[u[t, x, y] /. First[%]], {x, 0, inf}, {y, 0, inf}, 
  PlotRange -> All, PlotPoints -> 10, Mesh -> False], {t, 14, 15}]