differential equations – NDSolve does’t solve nonlinear system of PDEs with forth spatial derivative

I am trying to solve a highly nonlinear system of coupled PDEs with NDSolveValue.
In the following you can find the system, the boundary conditions and the value of the numerical parameters.

System of coupled PDEs

Boundary conditions

Numerical parameters

and here is my code

Numerical parameters

e = 1.60217662*10^-19;
h = 6.62607004*10^-34;
(CurlyPhi)0 = h/(2*Pi*2*e);
(CapitalPhi)0 = h/(2*e);
Ic = 4*10^-6;
Lj0 = h/(2*Pi*2*e*Ic);
Cj = 32*10^-15;
Cgp = 64*10^-15;
Cgs = 62*10^-15;
L = 160*10^-12;
Ej0 = (h*Ic)/(2*e*2*Pi);
k = 0.9;
(CapitalPhi)dc = (2*Pi*(CurlyPhi)0)/3;
(Omega)p = 2*Pi*10^9*12;
(Omega)s = 2*Pi*10^9*8;
a = 30*10^-6;

s = NDSolveValue(
{
   -Cj*(D(V(x, t) x, x, t, t) - Cgs/Cj*D(V(x, t), t, t))
     + (2*Ej0)/(CurlyPhi)0*(
       Sin(1/(CurlyPhi)0*(D(V(x, t), x) + (1/2)*D(V(x, t), x, x)))*
         Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) + 
            k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
                D(F(x, t), x, x)))) -
        Sin(1/(CurlyPhi)0*(D(V(x, t), x) - (1/2)*D(V(x, t), x, x)))*
         Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) + 
            k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
                D(F(x, t), x, x))))
       ) == 0,
   Cgp*D(F(x, t), t, t) - 1/L*D(F(x, t), x, x) +
     (k*Ej0)/(CurlyPhi)0*(
       Cos(1/(CurlyPhi)0*(D(V(x, t), x) - (1/2)*D(V(x, t), x, x)))*
         ((1/
             Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) + 
                k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
                    D(F(x, t), x, x))))) Cos((CapitalPhi)dc/(
             2*(CurlyPhi)0) + 
             k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
                 D(F(x, t), x, x)))*
           Sin((CapitalPhi)dc/(2*(CurlyPhi)0) + 
             k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
                 D(F(x, t), x, x))))
        - Cos(1/(CurlyPhi)0*(D(V(x, t), x) + (1/2)*D(V(x, t), x, x)))*
         ((1/
             Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) + 
                k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
                    D(F(x, t), x, x))))) Cos((CapitalPhi)dc/(
             2*(CurlyPhi)0) + 
             k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
                 D(F(x, t), x, x)))*
           Sin((CapitalPhi)dc/(2*(CurlyPhi)0) + 
             k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
                 D(F(x, t), x, x))))
       ) == 0,
   V(0, t) == ((CapitalPhi)0/20)*Sin((Omega)s*t),
   V(x, 0) == 0,
   F(0, t) == ((CapitalPhi)0/2)*Sin((Omega)p*t),
   F(x, 0) == 0
   },
  {F, V},
  {x, 0.003},
  {t, 10^-9}
  )

Now, every time I try to solve this system mathematica returns the following problem
enter image description here
Reading online I realized that this is a limit of mathematica 11.0 because my PDE contains a forth order derivative. For this reason I tried to change method of NDSolveValue using the following string

enter image description here

so the method of lines. But this time when I try to solve the system mathematica returns a different error message
enter image description here
This time I tried to change, add or remove some boundary conditions without really solving the problem.
Is there a different method I should try to solve this PDE system? Am I imposing wrong boundary conditions for the problem?
Thank you everybody for the help!

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enter image description here

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enter image description here

the parameter tao is as follows:

enter image description here

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enter image description here

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