# Hamilton Jacobi equation,

Let be the S function being the action in physics S=S(x,y,z,t) satisfying the equation:

$$\frac{dS}{dt}+(1/2m)(\nabla{S})^{2}+V(x,y,z,t)=0$$

where V is the potential is there any solution (exact) to it depending on V?

Implicit function theorem?

We can prove the existence and the unicity of the solution ,
What do you mean by$$\nabla{S})^{2}$$ is it the laplacian of S?
what is your boundary condition?is it Dirichlet or Neuman?

i m still wait your reponse

$$(\nabla S)^2=|\nabla S|^2 \qquad \hbox{the square gradient}$$

you can establish an equivalent system of ode's for your nonlinear problem and then aswer the questions for existence, unicity and solvability...
for more details check first chapter of Fritz John book.

In this case we can use the nonlinear semi group theory

dextercioby